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Old 2018-12-17, 21:13   #69
GP2
 
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Sep 2003

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Sorry if I'm hijacking this thread and chasing mirages, but... rather than looking at all fully-factored Mersenne numbers, let's look specifically at Mersenne semiprimes.

They seem to behave the opposite way from Mersenne primes.

Below is the list of the 325 currently-known fully-factored or probably-fully-factored Mersenne numbers. Alongside each exponent is a count of factors.

By convention, we always omit the cofactor from our factor lists even when it is a prime or probable prime. So the exponents listed below with a "1" next to them represent semi-primes, which strictly speaking have two factors.

Code:
      1 11
      1 23
      2 29
      1 37
      1 41
      2 43
      2 47
      2 53
      1 59
      1 67
      2 71
      2 73
      2 79
      1 83
      1 97
      1 101
      1 103
      1 109
      4 113
      1 131
      1 137
      1 139
      1 149
      4 151
      3 157
      4 163
      1 167
      3 173
      2 179
      3 181
      4 191
      2 193
      1 197
      1 199
      2 211
      5 223
      1 227
      3 229
      3 233
      5 239
      1 241
      4 251
      2 257
      3 263
      1 269
      1 271
      2 277
      1 281
      2 283
      1 293
      4 307
      2 311
      3 313
      3 317
      2 331
      4 337
      1 347
      2 349
      2 353
      5 359
      4 367
      1 373
      1 379
      4 383
      2 389
      8 397
      3 401
      2 409
      4 419
      1 421
      7 431
      3 433
      3 439
      2 443
      4 449
      1 457
      3 461
      5 463
      2 467
      3 479
      1 487
      6 491
      2 499
      3 503
      3 509
      1 523
      4 541
      3 547
      4 557
      2 563
      4 569
      3 571
      2 577
      5 587
      3 593
      2 599
      3 601
      2 613
      3 617
      3 619
      2 631
      5 641
      2 643
      2 647
      5 653
      3 659
      4 661
      3 673
      4 677
      2 683
      4 691
      7 701
      2 709
      3 719
      1 727
      3 733
      3 739
      6 743
      2 751
      3 757
      4 761
      2 769
      5 773
      3 787
      6 797
      1 809
      3 811
      4 821
      3 823
      5 827
      2 829
      4 839
      4 853
      3 857
      3 859
      4 863
      5 877
      1 881
      3 883
      4 887
      3 907
      3 911
      2 919
      4 929
      4 937
      2 941
      5 947
      5 953
      4 967
      1 971
      5 977
      1 983
      4 991
      1 997
      7 1009
      3 1013
      4 1019
      4 1021
      3 1031
      4 1033
      2 1039
      3 1049
      3 1051
      1 1061
      1 1063
      3 1069
      7 1087
      5 1091
      4 1093
      6 1097
      2 1103
      3 1109
      2 1117
      2 1123
      3 1129
      5 1151
      3 1153
      3 1163
      2 1171
      4 1181
      5 1187
      2 1193
      5 1201
      3 1223
      5 1289
      2 1301
      2 1303
      3 1307
      4 1321
      2 1327
      6 1361
      5 1373
      3 1409
      1 1427
      2 1459
      3 1471
      1 1487
      2 1531
      3 1543
      5 1553
      2 1559
      1 1637
      1 1657
      4 1693
      5 1783
      2 1907
      3 1997
      4 2069
      6 2087
      5 2243
      4 2251
      2 2311
      3 2381
      2 2383
      5 2447
      3 2549
      6 2677
      5 2699
      5 2789
      3 2837
      4 2909
      1 2927
      2 3041
      1 3079
      1 3259
      1 3359
      2 3547
      2 3833
      2 4127
      4 4219
      1 4243
      1 4729
      5 4751
      2 4871
      3 5087
      3 5227
      5 5233
      1 5689
      1 6043
      3 6199
      3 6337
      2 6883
      4 7039
      1 7331
      3 7417
      2 7673
      1 7757
      3 8243
      2 8849
      2 9697
      3 9733
      6 9901
      4 10007
      1 10169
      3 10211
      4 10433
      2 11117
      2 11813
      3 12451
      2 12569
      1 14561
      2 14621
      1 17029
      2 17683
      4 19121
      3 20521
      4 20887
      3 22193
      4 25243
      3 25933
      1 26903
      1 28759
      1 28771
      2 29473
      2 32531
      4 32611
      4 35339
      2 41263
      3 41521
      3 41681
      2 51487
      3 53381
      4 57131
      1 58199
      1 63703
      2 82939
      3 84211
      2 86137
      1 86371
      2 87691
      1 106391
      1 130439
      1 136883
      1 151013
      2 157457
      1 173867
      2 174533
      2 175631
      1 221509
      3 270059
      1 271211
      1 271549
      1 406583
      1 432457
      3 440399
      2 488441
      3 576551
      1 611999
      2 675977
      1 684127
      2 696343
      5 750151
      3 822971
      1 1010623
      1 1168183
      1 1304983
      1 1629469
      4 1790743
      1 2327417
      1 3464473
      1 4187251
      2 4834891
      1 5240707
      3 7080247
      1 7313983
If we look at only the exponents p of the 83 semiprimes we get the following:

Code:
             all   p>1200  p>10k  p>100k
     p=1      34     15     10      7
     p=3      49     29     20     14
So for exponents of Mersenne semiprimes there seems to be a marked prevalence of p=3 vs. p=1, whereas for exponents of Mersenne primes it is the opposite.

At higher values of p, the prevalence seems to be 2-to-1.

However, perhaps what is being selected for is not the size of p, but the asymmetricity of the two factors that make up the semiprime. In the Cunningham range below 1200, everything is fully-factored, but for higher exponents, we are only capable of finding very asymmetric semiprimes with one very small factor and an enormous cofactor.

Last fiddled with by GP2 on 2018-12-17 at 21:18
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