Table Of ContentA REFERENCE BOOK
FOR THE MECHANICAL ENGINEER, DESIGNER,
MANUFACTURING ENGINEER, DRAFTSMAN,
TOOLMAKER, AND MACHINIST
26th Edition
Machinery’s
Handbook
BY ERIK OBERG, FRANKLIN D. JONES,
HOLBROOK L. HORTON, AND HENRY H. RYFFEL
CHRISTOPHER J. MCCAULEY, EDITOR
RICCARDO HEALD, ASSOCIATE EDITOR
MUHAMMED IQBAL HUSSAIN, ASSOCIATE EDITOR
2000
INDUSTRIAL PRESS INC.
NEW YORK
COPYRIGHT 1914, 1924, 1928, 1930, 1931, 1934, 1936, 1937, 1939, 1940, 1941, 1942,
1943, 1944, 1945, 1946, 1948, 1950, 1951, 1952, 1953, 1954, 1955, 1956, 1957,© 1959, ©
1962, © 1964, © 1966, © 1968, © 1971, © 1974, © 1975, © 1977, © 1979, © 1984, ©
1988, © 1992, © 1996, © 1997, © 1998, © 2000 by Industrial Press Inc., New York, NY.
Library of Congress Cataloging-in-Publication Data
Oberg, Erik, 1881—1951
Machinery's Handbook.
2640 p.
Includes index.
I. Mechanical engineering—Handbook, manuals, etc.
I. Jones, Franklin Day, 1879-1967
II. Horton, Holbrook Lynedon, 1907-
III. Ryffel, Henry H. I920-
IV. Title.
TJ151.0245
2000
621.8'0212
72-622276
ISBN 0-8311-2625-6 (Thumb Indexed 11.7 x 17.8 cm)
ISBN 0-8311-2635-3 (Thumb Indexed 17.8 x 25.4 cm)
ISBN 0-8311-2666-3 (CD-ROM)
LC card number 72-622276
Printed and bound in the United States of America by National Publishing Company, Philadelphia,
Pa. All rights reserved. This book or parts thereof may not be reproduced, stored in a retrieval system,
or transmitted in any form without permission of the publishers.
INDUSTRIAL PRESS, INC.
200 Madison Avenue
New York, New York 10016-4078
MACHINERY'S HANDBOOK
26th Edition
First Printing
ACKNOWLEDGMENTS
The editors would like to acknowledge all those who contributed ideas and suggestions
to the Handbook. Several individuals and companies, in particular, contributed substantial
amounts of information to this edition.
Dr. Bertil Colding, Colding International Corp., Lansing, Michigan provided extensive
material on grinding speeds, feeds, depths of cut, and tool life for a wide range of materials.
He also provided practical information on machining econometrics, including tool wear
and tool life and machining cost relationships.
Dr. Bruce Harding, Director and Professor of MET, Purdue University contributed infor-
mation on GD&T as well material on the differences between ISO and ANSI surface finish
symbology.
Mr. Edward Craig of WeldTrain International and ABB Flexible Automation, Inc. con-
tributed information on pipe welding.
Mr. Sydney Kravitz provided valuable information on the weight of piles for different
materials.
Dr. T. A. Stolarski, Professor at Brunel University, provided detailed explanations and
equations on aerodynamic lubrication.
Mr. Alec Stokes provided much new material that was incorporated into gearing sections
including highpoint gears, British spur and helical gearing, addendum modification to
involute spur and helical gears, and hypoid gears.
Mr. Richard Pohanish contributed material on metal working fluids.
Hansvedt Industries provided a detailed listing of EDM terms.
Mr. Matthew Radcliff supplied data on wood screw pilot hole sizes.
Mr. Robert E. Green, as editor emeritus, contributed much useful, well organized mate-
rial to this edition. He also provided invaluable practical guidance to the editorial staff dur-
ing the Handbook’s compilation.
Finally, Industrial Press is extremely fortunate that Mr. Henry H. Ryffel, author and edi-
tor of Machinery’s Handbook, continues to be deeply involved with the Handbook.
Henry’s ideas, suggestions, and vision are deeply appreciated by everyone who worked on
this book.
PREFACE
v
PREFACE
For more than 85 years of continuous publication, Machinery's Handbook has served as
the principal reference in design and manufacturing facilities, and in colleges throughout
the world. The editors’ objective continues to be that of making the Handbook a practical
tool to be used in the same way that other kinds of tools are used, to make or repair products
of high quality, at the lowest cost, and in the shortest time possible.
Reference works such as Machinery's Handbook cannot carry the same information in
successive editions if they are to justify the claim that new or updated material is always
presented. The editors of such a book must move with the times, keeping a finger on the
pulse of manufacturing industry to learn what subjects have less, and what have more, use-
fulness to the majority of users. At the same time, material that is of proven worth must
continue to be included if the Handbook is to provide for the needs of disciplines that do not
develop as fast as, for instance, the numerical control field. Thus, it remains a difficult task
to select suitable material from the almost limitless supply of data pertaining to the manu-
facturing and mechanical engineering fields, and to provide for the needs of design and
production departments in all sizes of manufacturing plants and workshops, as well as
those of jobbing shops, trade schools, and technical schools.
The editors rely to some extent on conversations with users of the Handbook, and on
postcards and other written communications from Handbook users, for guidance on which
topics should be introduced, revised, lengthened, shortened, or omitted. In response to
users’ suggestions, in recent years material on logarithms, trigonometry, and other topics
was restored, and in this edition sine-bar tables have finally been restored after numerous
requests for this topic. Also at the request of users, in 1997 the first ever large-print or
“desktop” edition of the Handbook was published, followed in 1998 by the publication of
Machinery's Handbook CD-ROM including several hundred additional pages of material
restored from earlier editions.
Regular users of the Handbook should be able to identify some of the many changes
embodied in the present edition. “Old style” numerals, in continuous use since the first edi-
tion, and occasionally a source of confusion for readers, have been replaced by a modern
numeral style. The entire text of this edition, including all the tables and equations, has
been reset, and a great many of the numerous figures have been redrawn. The addition of
80 pages brings the total length of the book to 2640 pages.
The 26th edition of the Handbook contains significant format changes and major revi-
sions of existing content, as well as new material on a variety of topics including: aerody-
namic lubrication, high speed machining, grinding feeds and speeds, machining
econometrics, metalworking fluids, ISO surface texture, pipe welding, geometric dimen-
sioning and tolerancing, gearing, and EDM.
Other subjects in the Handbook that are new or have been revised, expanded, or updated
are: graphic descriptions of functions of angles, imaginary and complex numbers, com-
plex coordinate systems, contour milling, weight of piles, Ohm’s law, binary multiples,
force on inclined planes, and measurement over pins.
Those users involved in aspects of machining and grinding will be interested in the new
topics MACHINING ECONOMETRICS and GRINDING FEEDS AND SPEEDS, pre-
sented in the Machining section. The core of all manufacturing methods start with the cut-
ting edge and the metal removal process, and improving the control of the machining
process is a major component in order to achieve a Lean chain of manufacturing events.
These sections describe the means that are necessary to get metal cutting processes under
control and how to properly evaluate the decision making.
A major goal of the editors is to make the Handbook easier to use. The 26th edition of the
Handbook continues to incorporate the time-saving thumb tabs, much requested by users
in the past. The table of contents pages beginning each major section, first introduced for
the 25th edition, have proven very useful to readers. Consequently, the number of contents
pages has been increased to several pages each for many of the larger sections, more thor-
PREFACE
vi
oughly reflecting the contents of these sections. In the present edition, the Plastics section,
formerly a separate thumb tab, has been incorporated into the Properties of Materials sec-
tion. A major task in assembling this edition has been the expansion and reorganization of
the index. For the first time, most of the many Standards referenced in the Handbook are
now included in the index.
The American Standards Association was reconstituted in August 1969 as the United
States of America Standards Institute, and standards that had been approved as American
Standards were designated as USA Standards. In October 1969, the name was changed to
the American National Standards Institute. Thus, the designation of present standards is
ANSI instead of ASA or USAS. Standards originally adopted by the American Standards
Association and not revised are still referred to in the Handbook by the designation ASA.
ANSI Standards are copyrighted by the American National Standards Institute, West
42nd Street, New York, NY 10017, from whom current copies may be purchased. Many of
the American National Standards Institute (ANSI) Standards that deal with mechanical
engineering, extracts from which are included in the Handbook, are produced by the
American Society of Mechanical Engineers (ASME), and we are grateful for their permis-
sion to quote extracts and to update the information contained in the standards, based on
the revisions regularly carried out by the ASME. Information regarding current editions of
any of these Standards can be obtained from ASME International, Three Park Avenue,
New York, NY 10016.
Users who call possible defects to the attention of the editors, or the omission of some
matter that is considered to be of general value, often render a service to the entire manu-
facturing field. We desire to increase the usefulness of the Handbook as far as possible, so
all criticisms and suggestions about revisions, omissions, or inclusion of new material are
welcome.
Christopher J. McCauley, Editor
CONTENTS
Mathematics
1
Mechanics
112
Strength of Materials
193
Properties, Treatment, and Testing of Materials
361
Dimensioning, Gaging, and Measuring
605
Tooling and Toolmaking
720
Machining Operations
974
Manufacturing Processes
1298
Fasteners
1452
Threads and Threading
1702
Gears, Splines, and Cams
1995
Machine Elements
2189
Measuring Units
2513
Index
2556
TABLE OF CONTENTS
1
MATHEMATICS
NUMBERS, FRACTIONS, AND
DECIMALS
2 Mathematical Signs and
Commonly Used Abbreviations
3 Prime Numbers and Factors of
Numbers
13 Continued and Conjugate Fractions
16 Positive and Negative Numbers
17 Powers, Roots, and Reciprocals
17 Powers of Ten Notation
19 Preferred Numbers
ALGEBRA AND EQUATIONS
20 Rearrangement and Transposition
Formulas
21 Sequence of Performing Arithmetic
Operations
21 Ratio and Proportion
23
Inverse Proportion
23 Percentage
24
Simple and Compound Interest
25
Nominal vs. Effective Interest
Rates
32 Principal Algebraic Expressions
and Formulas
32 Equations
32
Solving First Degree Equations
33
Solving Quadratic Equations
33
Cubic Equations
33
Series
34
Derivatives of Functions
35
Solving Numerical Equations
36 Coordinate Systems
36
Rectangular, Cartesian
Coordinates
36
Polar Coordinates
36
Changing Coordinate Systems
37
Spherical Coordinates
38
Cylindrical Coordinates
38 Imaginary and Complex Numbers
38
Complex or Imaginary Numbers
39
Operations on Complex Numbers
40 Break-Even Analysis
GEOMETRY
43 Arithmetical & Geometrical
Progression
56 Areas and Volumes
56
The Prismoidal Formula
56
Pappus or Guldinus Rules
57
Area of Revolution Surface
57
Area of Irregular Plane Surface
58
Areas Enclosed by Cycloidal
Curves
58
Contents of Cylindrical Tanks
60 Areas and Dimensions of Figures
66 Volumes of Solids
72
Circles within Circles
77
Circles within Rectangles
78
Formulas for Regular Polygons
79 Diameters of Circles and Sides of
Squares of Equal Area
80
Segments of Circles
82
Distance Across Corners of
Squares and Hexagons
SOLUTION OF TRIANGLES
83 Functions of Angles
84
Laws of Sines and Cosines
84
Trigonometric Identities
90
Degree to Radian Conversion
table
92
Graphic Illustrations of the
Functions of Angles
93
Trigonometric Functions
97
Versed Sine and Versed Cosine
97
Sevolute and Involute Functions
LOGARITHMS
102
Common Logarithms
103
Inverse Logarithm
104
Natural Logarithms
104
Multiplication and Division by
Logarithms
106
Powers of Number by Logarithms
107
Roots of Number by Logarithms
108
Tables of Logarithms
2
MATHEMATICAL TABLES
NUMBERS, FRACTIONS, AND DECIMALS
Mathematical Signs and Commonly Used Abbreviations
+
Plus (sign of addition)
π
Pi (3.1416)
+
Positive
Σ
Sigma (sign of summation)
−
Minus (sign of subtraction)
ω
Omega (angles measured in radians)
−
Negative
g
Acceleration due to gravity (32.16 ft. per
sec. per sec.)
± (�)
Plus or minus (minus or plus)
i (or j)
Imaginary quantity
×
Multiplied by (multiplication sign)
sin
Sine
·
Multiplied by (multiplication sign)
cos
Cosine
÷
Divided by (division sign)
tan
Tangent
/
Divided by (division sign)
cot
Cotangent
:
Is to (in proportion)
sec
Secant
=
Equals
csc
Cosecant
≠
Is not equal to
vers
Versed sine
≡
Is identical to
covers
Coversed sine
::
Equals (in proportion)
sin−1 a
arcsin a
Arc the sine of which is a
≅
Approximately equals
(sin a)−1
Reciprocal of sin a (1 ÷ sin a)
≈
sinn x
nth power of sin x
>
Greater than
sinh x
Hyperbolic sine of x
<
Less than
cosh x
Hyperbolic cosine of x
≥
Greater than or equal to
∆
Delta (increment of)
≤
Less than or equal to
δ
Delta (variation of)
→
Approaches as a limit
d
Differential (in calculus)
�
Varies directly as
∂
Partial differentiation (in calculus)
∴
Therefore
∫
Integral (in calculus)
Square root
Integral between the limits a and b
Cube root
!
5! = 1 × 2 × 3 × 4 × 5 (Factorial)
4th root
∠
Angle
nth root1⁄2
�
Right angle
a2
a squared (2nd power of a)
⊥
Perpendicular to
a3
a cubed (3rd power of a)
�
Triangle
a4
4th power of a
�
Circle
an
nth power of a
�
Parallelogram
a−n
1 ÷ an
°
Degree (circular arc or temperature)
Reciprocal value of n
′
Minutes or feet
log
Logarithm
″
Seconds or inches
loge
Natural or Napierian logarithm
a′
a prime
ln
a″
a double prime
e
Base of natural logarithms (2.71828)
a1
a sub one
lim
Limit value (of an expression)
a2
a sub two
∞
Infinity
an
a sub n
α
Alpha
commonly used to
denote angles
( )
Parentheses
β
Beta
[ ]
Brackets
γ
Gamma
{ }
Braces
θ
Theta
φ
Phi
µ
Mu (coefficient of friction)
1
–
(
)
b
a∫
3
4
n
1
n--
FACTORS AND PRIME NUMBERS
3
Prime Numbers and Factors of Numbers
The factors of a given number are those numbers which when multiplied together give a
product equal to that number; thus, 2 and 3 are factors of 6; and 5 and 7 are factors of 35.
A prime number is one which has no factors except itself and 1. Thus, 3, 5, 7, 11, etc., are
prime numbers. A factor which is a prime number is called a prime factor.
The accompanying “Prime Number and Factor Tables” give the smallest prime factor of
all odd numbers from 1 to 9600, and can be used for finding all the factors for numbers up
to this limit. For example, find the factors of 931. In the column headed “900” and in the
line indicated by “31” in the left-hand column, the smallest prime factor is found to be 7. As
this leaves another factor 133 (since 931 ÷ 7 = 133), find the smallest prime factor of this
number. In the column headed “100” and in the line “33”, this is found to be 7, leaving a
factor 19. This latter is a prime number; hence, the factors of 931 are 7 × 7 × 19. Where no
factor is given for a number in the factor table, it indicates that the number is a prime num-
ber.
The last page of the tables lists all prime numbers from 9551 through 18691; and can be
used to identify quickly all unfactorable numbers in that range.
For factoring, the following general rules will be found useful:
2 is a factor of any number the right-hand figure of which is an even number or 0. Thus,
28 = 2 × 14, and 210 = 2 × 105.
3 is a factor of any number the sum of the figures of which is evenly divisible by 3. Thus,
3 is a factor of 1869, because 1 + 8 + 6 + 9 = 24 ÷ 3 = 8.
4 is a factor of any number the two right-hand figures of which, considered as one num-
ber, are evenly divisible by 4. Thus, 1844 has a factor 4, because 44 ÷ 4 = 11.
5 is a factor of any number the right-hand figure of which is 0 or 5. Thus, 85 = 5 × 17; 70
= 5 × 14.
Tables of prime numbers and factors of numbers are particularly useful for calculations
involving change-gear ratios for compound gearing, dividing heads, gear-generating
machines, and mechanical designs having gear trains.
Example 1:A set of four gears is required in a mechanical design to provide an overall
gear ratio of 4104 ÷ 1200. Furthermore, no gear in the set is to have more than 120 teeth or
less than 24 teeth. Determine the tooth numbers.
First, as explained previously, the factors of 4104 are determined to be: 2 × 2 × 2 × 3 × 3
× 57 = 4104. Next, the factors of 1200 are determined: 2 × 2 × 2 × 2 × 5 × 5 × 3 = 1200.
Therefore
. If the factors had been com-
bined differently, say, to give
, then the 16-tooth gear in the denominator would
not satisfy the requirement of no less than 24 teeth.
Example 2:Factor the number 25078 into two numbers neither of which is larger than
200.
The first factor of 25078 is obviously 2, leaving 25078 ÷ 2 = 12539 to be factored further.
However, from the last table, Prime Numbers from 9551 to 18691, it is seen that 12539 is a
prime number; therefore, no solution exists.
4104
1200
------------
2
2
×
2
×
3
×
3
×
57
×
2
2
×
2
×
2
×
5
×
5
×
3
×
----------------------------------------------------------
72
57
×
24
50
×
------------------
=
=
72
57
×
16
75
×
------------------
4
FACTORS AND PRIME NUMBERS
Prime Number and Factor Table for 1 to 1199
From
To
0
100
100
200
200
300
300
400
400
500
500
600
600
700
700
800
800
900
900
1000
1000
1100
1100
1200
1
P
P
3
7
P
3
P
P
3
17
7
3
3
P
P
7
3
13
P
3
19
11
3
17
P
5
P
3
5
5
3
5
5
3
5
5
3
5
7
P
P
3
P
11
3
P
7
3
P
19
3
9
3
P
11
3
P
P
3
P
P
3
P
P
11
P
3
P
P
3
7
13
3
P
P
3
11
13
P
P
3
P
7
3
P
23
3
11
P
3
15
3
5
5
3
5
5
3
5
5
3
5
5
17
P
3
7
P
3
11
P
3
19
7
3
P
19
P
7
3
11
P
3
P
P
3
P
P
3
21
3
11
13
3
P
P
3
7
P
3
P
19
23
P
3
P
17
3
P
7
3
P
13
3
P
25
5
5
3
5
5
3
5
5
3
5
5
3
27
3
P
P
3
7
17
3
P
P
3
13
7
29
P
3
P
7
3
23
17
3
P
P
3
P
31
P
P
3
P
P
3
P
17
3
7
P
3
33
3
7
P
3
P
13
3
P
7
3
P
11
35
5
3
5
5
3
5
5
3
5
5
3
5
37
P
P
3
P
19
3
7
11
3
P
17
3
39
3
P
P
3
P
7
3
P
P
3
P
17
41
P
3
P
11
3
P
P
3
29
P
3
7
43
P
11
3
7
P
3
P
P
3
23
7
3
45
3
5
5
3
5
5
3
5
5
3
5
5
47
P
3
13
P
3
P
P
3
7
P
3
31
49
7
P
3
P
P
3
11
7
3
13
P
3
51
3
P
P
3
11
19
3
P
23
3
P
P
53
P
3
11
P
3
7
P
3
P
P
3
P
55
5
5
3
5
5
3
5
5
3
5
5
3
57
3
P
P
3
P
P
3
P
P
3
7
13
59
P
3
7
P
3
13
P
3
P
7
3
19
61
P
7
3
19
P
3
P
P
3
31
P
3
63
3
P
P
3
P
P
3
7
P
3
P
P
65
5
3
5
5
3
5
5
3
5
5
3
5
67
P
P
3
P
P
3
23
13
3
P
11
3
69
3
13
P
3
7
P
3
P
11
3
P
7
71
P
3
P
7
3
P
11
3
13
P
3
P
73
P
P
3
P
11
3
P
P
3
7
29
3
75
3
5
5
3
5
5
3
5
5
3
5
5
77
7
3
P
13
3
P
P
3
P
P
3
11
79
P
P
3
P
P
3
7
19
3
11
13
3
81
3
P
P
3
13
7
3
11
P
3
23
P
83
P
3
P
P
3
11
P
3
P
P
3
7
85
5
5
3
5
5
3
5
5
3
5
5
3
87
3
11
7
3
P
P
3
P
P
3
P
P
89
P
3
17
P
3
19
13
3
7
23
3
29
91
7
P
3
17
P
3
P
7
3
P
P
3
93
3
P
P
3
17
P
3
13
19
3
P
P
95
5
3
5
5
3
5
5
3
5
5
3
5
97
P
P
3
P
7
3
17
P
3
P
P
3
99
3
P
13
3
P
P
3
17
29
3
7
11
FACTORS AND PRIME NUMBERS
5
Prime Number and Factor Table for 1201 to 2399
From
To
1200
1300
1300
1400
1400
1500
1500
1600
1600
1700
1700
1800
1800
1900
1900
2000
2000
2100
2100
2200
2200
2300
2300
2400
1
P
P
3
19
P
3
P
P
3
11
31
3
3
3
P
23
3
7
13
3
11
P
3
P
7
5
5
3
5
5
3
5
5
3
5
5
3
5
7
17
P
3
11
P
3
13
P
3
7
P
3
9
3
7
P
3
P
P
3
23
7
3
47
P
11
7
3
17
P
3
29
P
3
P
P
3
P
13
P
13
3
17
P
3
7
P
3
P
P
3
15
3
5
5
3
5
5
3
5
5
3
5
5
17
P
3
13
37
3
17
23
3
P
29
3
7
19
23
P
3
7
P
3
17
19
3
13
7
3
21
3
P
7
3
P
P
3
17
43
3
P
11
23
P
3
P
P
3
P
P
3
7
11
3
23
25
5
5
3
5
5
3
5
5
3
5
5
3
27
3
P
P
3
P
11
3
41
P
3
17
13
29
P
3
P
11
3
7
31
3
P
P
3
17
31
P
11
3
P
7
3
P
P
3
P
23
3
33
3
31
P
3
23
P
3
P
19
3
7
P
35
5
3
5
5
3
5
5
3
5
5
3
5
37
P
7
3
29
P
3
11
13
3
P
P
3
39
3
13
P
3
11
37
3
7
P
3
P
P
41
17
3
11
23
3
P
7
3
13
P
3
P
43
11
17
3
P
31
3
19
29
3
P
P
3
45
3
5
5
3
5
5
3
5
5
3
5
5
47
29
3
P
7
3
P
P
3
23
19
3
P
49
P
19
3
P
17
3
43
P
3
7
13
3
51
3
7
P
3
13
17
3
P
7
3
P
P
53
7
3
P
P
3
P
17
3
P
P
3
13
55
5
5
3
5
5
3
5
5
3
5
5
3
57
3
23
31
3
P
7
3
19
11
3
37
P
59
P
3
P
P
3
P
11
3
29
17
3
7
61
13
P
3
7
11
3
P
37
3
P
7
3
63
3
29
7
3
P
41
3
13
P
3
31
17
65
5
3
5
5
3
5
5
3
5
5
3
5
67
7
P
3
P
P
3
P
7
3
11
P
3
69
3
37
13
3
P
29
3
11
P
3
P
23
71
31
3
P
P
3
7
P
3
19
13
3
P
73
19
P
3
11
7
3
P
P
3
41
P
3
75
3
5
5
3
5
5
3
5
5
3
5
5
77
P
3
7
19
3
P
P
3
31
7
3
P
79
P
7
3
P
23
3
P
P
3
P
43
3
81
3
P
P
3
41
13
3
7
P
3
P
P
83
P
3
P
P
3
P
7
3
P
37
3
P
85
5
5
3
5
5
3
5
5
3
5
5
3
87
3
19
P
3
7
P
3
P
P
3
P
7
89
P
3
P
7
3
P
P
3
P
11
3
P
91
P
13
3
37
19
3
31
11
3
7
29
3
93
3
7
P
3
P
11
3
P
7
3
P
P
95
5
3
5
5
3
5
5
3
5
5
3
5
97
P
11
3
P
P
3
7
P
3
13
P
3
99
3
P
P
3
P
7
3
P
P
3
11
P
6
FACTORS AND PRIME NUMBERS
Prime Number and Factor Table for 2401 to 3599
From
To
2400
2500
2500
2600
2600
2700
2700
2800
2800
2900
2900
3000
3000
3100
3100
3200
3200
3300
3300
3400
3400
3500
3500
3600
1
7
41
3
37
P
3
P
7
3
P
19
3
3
3
P
19
3
P
P
3
29
P
3
41
31
5
5
3
5
5
3
5
5
3
5
5
3
5
7
29
23
3
P
7
3
31
13
3
P
P
3
9
3
13
P
3
53
P
3
P
P
3
7
11
11
P
3
7
P
3
41
P
3
13
7
3
P
13
19
7
3
P
29
3
23
11
3
P
P
3
15
3
5
5
3
5
5
3
5
5
3
5
5
17
P
3
P
11
3
P
7
3
P
31
3
P
19
41
11
3
P
P
3
P
P
3
P
13
3
21
3
P
P
3
7
23
3
P
P
3
11
7
23
P
3
43
7
3
37
P
3
11
P
3
13
25
5
5
3
5
5
3
5
5
3
5
5
3
27
3
7
37
3
11
P
3
53
7
3
23
P
29
7
3
11
P
3
29
13
3
P
P
3
P
31
11
P
3
P
19
3
7
31
3
P
47
3
33
3
17
P
3
P
7
3
13
53
3
P
P
35
5
3
5
5
3
5
5
3
5
5
3
5
37
P
43
3
7
P
3
P
P
3
47
7
3
39
3
P
7
3
17
P
3
43
41
3
19
P
41
P
3
19
P
3
17
P
3
7
13
3
P
43
7
P
3
13
P
3
17
7
3
P
11
3
45
3
5
5
3
5
5
3
5
5
3
5
5
47
P
3
P
41
3
7
11
3
17
P
3
P
49
31
P
3
P
7
3
P
47
3
17
P
3
51
3
P
11
3
P
13
3
23
P
3
7
53
53
11
3
7
P
3
P
43
3
P
7
3
11
55
5
5
3
5
5
3
5
5
3
5
5
3
57
3
P
P
3
P
P
3
7
P
3
P
P
59
P
3
P
31
3
11
7
3
P
P
3
P
61
23
13
3
11
P
3
P
29
3
P
P
3
63
3
11
P
3
7
P
3
P
13
3
P
7
65
5
3
5
5
3
5
5
3
5
5
3
5
67
P
17
3
P
47
3
P
P
3
7
P
3
69
3
7
17
3
19
P
3
P
7
3
P
43
71
7
3
P
17
3
P
37
3
P
P
3
P
73
P
31
3
47
13
3
7
19
3
P
23
3
75
3
5
5
3
5
5
3
5
5
3
5
5
77
P
3
P
P
3
13
17
3
29
11
3
7
79
37
P
3
7
P
3
P
11
3
31
7
3
81
3
29
7
3
43
11
3
P
17
3
59
P
83
13
3
P
11
3
19
P
3
7
17
3
P
85
5
5
3
5
5
3
5
5
3
5
5
3
87
3
13
P
3
P
29
3
P
19
3
11
17
89
19
3
P
P
3
7
P
3
11
P
3
37
91
47
P
3
P
7
3
11
P
3
P
P
3
93
3
P
P
3
11
41
3
31
37
3
7
P
95
5
3
5
5
3
5
5
3
5
5
3
5
97
11
7
3
P
P
3
19
23
3
43
13
3
99
3
23
P
3
13
P
3
7
P
3
P
59
FACTORS AND PRIME NUMBERS
7
Prime Number and Factor Table for 3601 to 4799
From
To
3600
3700
3700
3800
3800
3900
3900
4000
4000
4100
4100
4200
4200
4300
4300
4400
4400
4500
4500
4600
4600
4700
4700
4800
1
13
P
3
47
P
3
P
11
3
7
43
3
3
3
7
P
3
P
11
3
13
7
3
P
P
5
5
3
5
5
3
5
5
3
5
5
3
5
7
P
11
3
P
P
3
7
59
3
P
17
3
9
3
P
13
3
19
7
3
31
P
3
11
17
11
23
3
37
P
3
P
P
3
11
13
3
7
13
P
47
3
7
P
3
11
19
3
P
7
3
15
3
5
5
3
5
5
3
5
5
3
5
5
17
P
3
11
P
3
23
P
3
7
P
3
53
19
7
P
3
P
P
3
P
7
3
P
31
3
21
3
61
P
3
P
13
3
29
P
3
P
P
23
P
3
P
P
3
7
41
3
P
P
3
P
25
5
5
3
5
5
3
5
5
3
5
5
3
27
3
P
43
3
P
P
3
P
19
3
7
29
29
19
3
7
P
3
P
P
3
43
7
3
P
31
P
7
3
P
29
3
P
61
3
23
11
3
33
3
P
P
3
37
P
3
7
11
3
41
P
35
5
3
5
5
3
5
5
3
5
5
3
5
37
P
37
3
31
11
3
19
P
3
13
P
3
39
3
P
11
3
7
P
3
P
23
3
P
7
41
11
3
23
7
3
41
P
3
P
19
3
11
43
P
19
3
P
13
3
P
43
3
7
P
3
45
3
5
5
3
5
5
3
5
5
3
5
5
47
7
3
P
P
3
11
31
3
P
P
3
47
49
41
23
3
11
P
3
7
P
3
P
P
3
51
3
11
P
3
P
7
3
19
P
3
P
P
53
13
3
P
59
3
P
P
3
61
29
3
7
55
5
5
3
5
5
3
5
5
3
5
5
3
57
3
13
7
3
P
P
3
P
P
3
P
67
59
P
3
17
37
3
P
P
3
7
47
3
P
61
7
P
3
17
31
3
P
7
3
P
59
3
63
3
53
P
3
17
23
3
P
P
3
P
11
65
5
3
5
5
3
5
5
3
5
5
3
5
67
19
P
3
P
7
3
17
11
3
P
13
3
69
3
P
53
3
13
11
3
17
41
3
7
19
71
P
3
7
11
3
43
P
3
17
7
3
13
73
P
7
3
29
P
3
P
P
3
17
P
3
75
3
5
5
3
5
5
3
5
5
3
5
5
77
P
3
P
41
3
P
7
3
11
23
3
17
79
13
P
3
23
P
3
11
29
3
19
P
3
81
3
19
P
3
7
37
3
13
P
3
31
7
83
29
3
11
7
3
47
P
3
P
P
3
P
85
5
5
3
5
5
3
5
5
3
5
5
3
87
3
7
13
3
61
53
3
41
7
3
43
P
89
7
3
P
P
3
59
P
3
67
13
3
P
91
P
17
3
13
P
3
7
P
3
P
P
3
93
3
P
17
3
P
7
3
23
P
3
13
P
95
5
3
5
5
3
5
5
3
5
5
3
5
97
P
P
3
7
17
3
P
P
3
P
7
3
99
3
29
7
3
P
13
3
53
11
3
37
P
8
FACTORS AND PRIME NUMBERS
Prime Number and Factor Table for 4801 to 5999
From
To
4800
4900
4900
5000
5000
5100
5100
5200
5200
5300
5300
5400
5400
5500
5500
5600
5600
5700
5700
5800
5800
5900
5900
6000
1
P
13
3
P
7
3
11
P
3
P
P
3
3
3
P
P
3
11
P
3
P
13
3
7
P
5
5
3
5
5
3
5
5
3
5
5
3
5
7
11
7
3
P
41
3
P
P
3
13
P
3
9
3
P
P
3
P
P
3
7
71
3
37
19
11
17
3
P
19
3
47
7
3
31
P
3
23
13
P
17
3
P
13
3
P
37
3
29
P
3
15
3
5
5
3
5
5
3
5
5
3
5
5
17
P
3
29
7
3
13
P
3
41
P
3
61
19
61
P
3
P
17
3
P
P
3
7
11
3
21
3
7
P
3
23
17
3
P
7
3
P
31
23
7
3
P
47
3
P
11
3
P
59
3
P
25
5
5
3
5
5
3
5
5
3
5
5
3
27
3
13
11
3
P
7
3
P
17
3
P
P
29
11
3
47
23
3
73
61
3
13
17
3
7
31
P
P
3
7
P
3
P
P
3
11
7
3
33
3
P
7
3
P
P
3
11
43
3
19
17
35
5
3
5
5
3
5
5
3
5
5
3
5
37
7
P
3
11
P
3
P
7
3
P
13
3
39
3
11
P
3
13
19
3
29
P
3
P
P
41
47
3
71
53
3
7
P
3
P
P
3
13
43
29
P
3
37
7
3
P
23
3
P
P
3
45
3
5
5
3
5
5
3
5
5
3
5
5
47
37
3
7
P
3
P
13
3
P
7
3
19
49
13
7
3
19
29
3
P
31
3
P
P
3
51
3
P
P
3
59
P
3
7
P
3
P
11
53
23
3
31
P
3
53
7
3
P
11
3
P
55
5
5
3
5
5
3
5
5
3
5
5
3
57
3
P
13
3
7
11
3
P
P
3
P
7
59
43
3
P
7
3
23
53
3
P
13
3
59
61
P
11
3
13
P
3
43
67
3
7
P
3
63
3
7
61
3
19
31
3
P
7
3
11
67
65
5
3
5
5
3
5
5
3
5
5
3
5
67
31
P
3
P
23
3
7
19
3
73
P
3
69
3
P
37
3
11
7
3
P
P
3
P
47
71
P
3
11
P
3
41
P
3
53
29
3
7
73
11
P
3
7
P
3
13
P
3
23
7
3
75
3
5
5
3
5
5
3
5
5
3
5
5
77
P
3
P
31
3
19
P
3
7
53
3
43
79
7
13
3
P
P
3
P
7
3
P
P
3
81
3
17
P
3
P
P
3
P
13
3
P
P
83
19
3
13
71
3
7
P
3
P
P
3
31
85
5
5
3
5
5
3
5
5
3
5
5
3
87
3
P
P
3
17
P
3
37
11
3
7
P
89
P
3
7
P
3
17
11
3
P
7
3
53
91
67
7
3
29
11
3
17
P
3
P
43
3
93
3
P
11
3
67
P
3
7
P
3
71
13
95
5
3
5
5
3
5
5
3
5
5
3
5
97
59
19
3
P
P
3
23
29
3
11
P
3
99
3
P
P
3
7
P
3
11
41
3
17
7
FACTORS AND PRIME NUMBERS
9
Prime Number and Factor Table for 6001 to 7199
From
To
6000
6100
6100
6200
6200
6300
6300
6400
6400
6500
6500
6600
6600
6700
6700
6800
6800
6900
6900
7000
7000
7100
7100
7200
1
17
P
3
P
37
3
7
P
3
67
P
3
3
3
17
P
3
19
7
3
P
P
3
47
P
5
5
3
5
5
3
5
5
3
5
5
3
5
7
P
31
3
7
43
3
P
19
3
P
7
3
9
3
41
7
3
13
23
3
P
11
3
43
P
11
P
3
P
P
3
17
11
3
7
P
3
13
13
7
P
3
59
11
3
17
7
3
31
P
3
15
3
5
5
3
5
5
3
5
5
3
5
5
17
11
3
P
P
3
7
13
3
17
P
3
11
19
13
29
3
71
7
3
P
P
3
11
P
3
21
3
P
P
3
P
P
3
11
19
3
7
P
23
19
3
7
P
3
11
37
3
P
7
3
17
25
5
5
3
5
5
3
5
5
3
5
5
3
27
3
11
13
3
P
61
3
7
P
3
P
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