Small range with high density of factors
 

By accident I noticed a high occurrence of factors (59+60 bit) in the 37M range. 6 factors within 17 exponents or 12 factors within 155 exponents have been found (expected average is about 5 factors).
Very often I've seen 2 or 3 factors close together (also during P-1 factoring) but never that much. I guess there are much more of such hot spots. OTH, there are big ranges with no factors found. Is there any mathematical theory available for that phenomenon?

[CODE]
M37636561 has a factor: 1135828231306295921
M37636591 no factor to 2^60, WZ1: 6947FA77
M37636609 no factor to 2^60, WZ1: 693CFA7E
M37636619 has a factor: 511024828036639799
M37636633 has a factor: 570319236002859601
M37636673 no factor to 2^60, WZ1: 6942FA7C
M37636699 no factor to 2^60, WZ1: 693FFA76
M37636847 has a factor: 631878724251180671
M37636897 no factor to 2^60, WZ1: 693AFA76
M37636913 no factor to 2^60, WZ1: 694AFA7B
M37636981 no factor to 2^60, WZ1: 6955FA7D
M37636999 has a factor: 597659352683030201
M37637053 no factor to 2^60, WZ1: 6946FA78
M37637087 no factor to 2^60, WZ1: 694BFA79
M37637137 no factor to 2^60, WZ1: 6943FA7F
M37637227 no factor to 2^60, WZ1: 6946FA76
M37637287 has a factor: 405855766653864991
M37637359 no factor to 2^60, WZ1: 6957FA77
M37637377 no factor to 2^60, WZ1: 694CFA7E
M37637381 no factor to 2^60, WZ1: 6950FA77
M37637419 no factor to 2^60, WZ1: 693CFA7C
M37637437 no factor to 2^60, WZ1: 694EFA78
M37637491 no factor to 2^60, WZ1: 694AFA77
M37637519 no factor to 2^60, WZ1: 6949FA7D
M37637521 no factor to 2^60, WZ1: 694BFA7F
M37637557 no factor to 2^60, WZ1: 6952FA77
M37637573 no factor to 2^60, WZ1: 6945FA7C
M37637581 has a factor: 407576741069557993
.
.
.
M37638347 has a factor: 496922260335323759
.
M37639141 has a factor: 692248891302635353
.
M37640527 has a factor: 394094644116888527
.
M37641883 has a factor: 1048689918573638239
.
M37642147 has a factor: 522097453166506223
[/CODE]
Another example (6 factors within 19 exponents tested) :
[CODE]M37094417 has a factor: 345130813545920327
.
M37095103 has a factor: 687749134153090337
M37095131 no factor to 2^60, WZ1: 6413F6E1
M37095193 no factor to 2^60, WZ1: 6417F6DD
M37095199 no factor to 2^60, WZ1: 641DF6E3
M37095203 has a factor: 294807165076037863
M37095217 no factor to 2^60, WZ1: 6413F6DF
M37095329 no factor to 2^60, WZ1: 640FF6E1
M37095403 no factor to 2^60, WZ1: 641FF6DE
M37095451 no factor to 2^60, WZ1: 6415F6E2
M37095493 no factor to 2^60, WZ1: 6422F6E1
M37095517 no factor to 2^60, WZ1: 641DF6E3
M37095587 has a factor: 1105854523979714329
M37095593 has a factor: 401202455567851999
M37095841 no factor to 2^60, WZ1: 6423F6DD
M37095889 has a factor: 680303185401949913
M37095893 no factor to 2^60, WZ1: 641DF6E5
M37095967 no factor to 2^60, WZ1: 6410F6E2
M37095977 no factor to 2^60, WZ1: 641AF6E1
M37096013 has a factor: 1014310337932612439
.
M37096457 has a factor: 644196952627581481
[/CODE]

[QUOTE=hbock]12 factors within 155 exponents have been found (expected average is about 5 factors) ... Is there any mathematical theory available for that phenomenon?[/QUOTE]

The expected frequency of clusters can be calculated from the Poisson Approximation. I attempted to describe this as part of the [URL=http://home.earthlink.net/~elevensmooth/MathFAQ.html#Poisson]ElevenSmooth FAQ[/URL]. The probability of finding at least 12 events in a space where, on average, you expect to find 5. is about 5 in a thousand. Counting the number of spaces is tricky because of overlaps, but if you divided the range into groups of 155 exponents before ever looking at them, you would expect about 1 in 200 ranges to have at least 12 factors.

William