Factorization Tables

Introduction

As part of my research into multiperfect numbers and harmonic numbers, there is a need to have factorizations of the sum of divisor function, sigma(N). Since sigma(N) is a multiplicative function, the calculation of sigma(N) involves the calculation (or table look-up) of sigma() of appropriate prime powers. In particular the sigma() of small primes to large powers can be needed. Tables of such factorizations exist as the ongoing Cunningham project (Sam Wagstaff and Richard Brent) and are regularly updated. This is my modest attempt at making such information, together with my own extensions, easily available.

The following are some of the accumulated (Cunningham/Brent) factorizations used in my research.

Factorization of p^a+1
Factorization of p^a-1

and some of those resulting from my research

Factorization of sigma(p^a)
Factorization cascade for sigma(p^a)

Table Limits

The upper limits on the exponent a for various prime p are as follows:

p	p^a+1	p^a-1	sigma(p^a) chain
2	500	995	995
3	250	499	499
5	150	297	297
7	100	199	199
11	100	195	195
13	100	195	195
17	100	195	195
19	100	177	177
23	100	179	179
29	100	171	171
31	50	99	99
37	50	99	99
41-97	30	59	59
101-397	25	49	49
401-997	20	39	39
1009-1999	10	19	n.a.
2003-9973	10	19	n.a.

Notes:

Only odd exponents are listed for p^a-1 since even exponents can be found using the identity p^2a-1 = (p^a+1)(p^a-1)
NOTE...NOTE the sigma factor chain for 3^141 and 3^205 (and consequently 3^283), all the other chains are built of the assumption that any prime factor >10^6 will occur with exponent 1 only. (See sigma chains for more information.)

Missing p^a-1 factorizations
p last a missing a

2- 1185 641,643,671,673,679,683,689,713,719,727,731,739,743,751,757,
761,763,773,779,781,785,787,791,793,797,799,805,809,811,815,817,
821,823,827,841,845,853,857,859,863,877,887,893,895,899,905,
913,919,923,929,933,935,937,939,941,947,949,953,957,959,961,
963,967,971,973,977,985,991,997,999

3- 497 379,389,397,431,437,443,449,451,461,469,479,481,487,493

11- 197 191

13- 195 139,151,161,167,181,193

17- 195 103,107,109,121,129,137,139,143,149,151,155,167,169,173,181,193

19- 199 129,139,147,155,163,169,173,179,183,187,189,191,193,195,197

23- 187 101,109,113,115,131,139,143,149,155,157,167,173,175,181,183,185

29- 171 101,113,121,127,149,155,161,163,167,169

Missing p^a-1 factorizations
p	last a	missing a
2-	1185	641,643,671,673,679,683,689,713,719,727,731,739,743,751,757, 761,763,773,779,781,785,787,791,793,797,799,805,809,811,815,817, 821,823,827,841,845,853,857,859,863,877,887,893,895,899,905, 913,919,923,929,933,935,937,939,941,947,949,953,957,959,961, 963,967,971,973,977,985,991,997,999
3-	497	379,389,397,431,437,443,449,451,461,469,479,481,487,493
11-	197	191
13-	195	139,151,161,167,181,193
17-	195	103,107,109,121,129,137,139,143,149,151,155,167,169,173,181,193
19-	199	129,139,147,155,163,169,173,179,183,187,189,191,193,195,197
23-	187	101,109,113,115,131,139,143,149,155,157,167,173,175,181,183,185
29-	171	101,113,121,127,149,155,161,163,167,169

Missing sigma(p^a) chains
p last a missing a

2 999 532,546,640,642,646,670,672,678,682,688,700,
712,718,720,726,730,738,742,750,754,756,760,762,768,772,778,
780,784,786,790,792,796,798,804,808,810,814,816,820,822,826,828,
838,840,844,846,852,856,858,862,864,876,882,886,888,892,894,
895,896,898,900,904,906,910,912,918,922,924,928,932,934,936,938,940,
946,948,952,956,958,960,962,964,966,970,972,976,978,982,984,
988,990,992,996,998

3 499 276,286,336,352,368,378,388,396,402,430,436,438,442,448,450,460,466,
468,478,480,486,492,498

5 299 196,222,238,268,270,276,278,280,286,298

7 199 156,178,180,196

11 195 160,162,172,174,180,190,192

Missing sigma(p^a) chains
p	last a	missing a
2	999	532,546,640,642,646,670,672,678,682,688,700, 712,718,720,726,730,738,742,750,754,756,760,762,768,772,778, 780,784,786,790,792,796,798,804,808,810,814,816,820,822,826,828, 838,840,844,846,852,856,858,862,864,876,882,886,888,892,894, 895,896,898,900,904,906,910,912,918,922,924,928,932,934,936,938,940, 946,948,952,956,958,960,962,964,966,970,972,976,978,982,984, 988,990,992,996,998
3	499	276,286,336,352,368,378,388,396,402,430,436,438,442,448,450,460,466, 468,478,480,486,492,498
5	299	196,222,238,268,270,276,278,280,286,298
7	199	156,178,180,196
11	195	160,162,172,174,180,190,192

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Recent Changes

Contact the Author

Please send any comments, corrections or additions to
Ronald M. Sorli,
School of Mathematical Sciences,
University of Technology, Sydney
Australia

email: rons@maths.uts.edu.au

Disclaimer

While every care has been taken in assembling these tables, no guarantee as to the correctness of the values is given. Any errors are the sole responsibility of this author.

Last updated on 29 March 2001