[seqfan] Re: zig-zg pseudoprimes

[Richard Mathar]

http://list.seqfan.eu/pipermail/seqfan/2010-September/005915.html claims

vs>   I think, nevertheless, that your conclusion about the statistics of sequence {n: A000111(n)==1
vs> (mod n)}, maybe, is not true for large n. Indeed, the 92 first terms of this sequence which you presented one can split into 4 subsequences: 1) odd primes; 2) positive powers of 2; 3) numbers of the form 2*p, where p is an odd prime; 4)  remaider subsequence {30,182,...}.
vs> Till now I proved that  1) and 2) are indeed subsequences of the considered sequence and it is very plausible ( I did not yet prove this) that 3) is also subsequence. Since, nevertheless, union of these sequences has zero density, then your statement is true only in case when sequence {30,182,...} has a large density. It seems improbable.

Here is a binning statistics of the index range n, then the count
of A000111 in that index range with A000111(n) == 1( mod n):

range 1 .. 20 : 10  out of 20
range 21 .. 40 : 8  out of 20
range 41 .. 60 : 4  out of 20
range 61 .. 80 : 5  out of 20
range 81 .. 100 : 5  out of 20
range 101 .. 120 : 5  out of 20
range 121 .. 140 : 4  out of 20
range 141 .. 160 : 5  out of 20
range 161 .. 180 : 3  out of 20
range 181 .. 200 : 5  out of 20
range 201 .. 220 : 4  out of 20
range 221 .. 240 : 3  out of 20
range 241 .. 260 : 4  out of 20
range 261 .. 280 : 5  out of 20
range 281 .. 300 : 3  out of 20
range 301 .. 320 : 4  out of 20
range 321 .. 340 : 3  out of 20
range 341 .. 360 : 4  out of 20
range 361 .. 380 : 2  out of 20
range 381 .. 400 : 6  out of 20
range 401 .. 420 : 2  out of 20
range 421 .. 440 : 3  out of 20
range 441 .. 460 : 5  out of 20
range 461 .. 480 : 3  out of 20
range 481 .. 500 : 1  out of 20
range 501 .. 520 : 4  out of 20
range 521 .. 540 : 3  out of 20
range 541 .. 560 : 4  out of 20
range 561 .. 580 : 6  out of 20
range 581 .. 600 : 2  out of 20
range 601 .. 620 : 4  out of 20
range 621 .. 640 : 4  out of 20
range 641 .. 660 : 2  out of 20
range 661 .. 680 : 5  out of 20
range 681 .. 700 : 2  out of 20
range 701 .. 720 : 4  out of 20
range 721 .. 740 : 2  out of 20
range 741 .. 760 : 3  out of 20
range 761 .. 780 : 5  out of 20
range 781 .. 800 : 3  out of 20
range 801 .. 820 : 3  out of 20
range 821 .. 840 : 3  out of 20
range 841 .. 860 : 4  out of 20
range 861 .. 880 : 5  out of 20
range 881 .. 900 : 3  out of 20
range 901 .. 920 : 1  out of 20
range 921 .. 940 : 5  out of 20
range 941 .. 960 : 3  out of 20
range 961 .. 980 : 2  out of 20
range 981 .. 1000 : 3  out of 20
range 1001 .. 1020 : 4  out of 20
range 1021 .. 1040 : 3  out of 20
range 1041 .. 1060 : 3  out of 20
range 1061 .. 1080 : 2  out of 20
range 1081 .. 1100 : 4  out of 20
range 1101 .. 1120 : 4  out of 20
range 1121 .. 1140 : 4  out of 20
range 1141 .. 1160 : 3  out of 20
range 1161 .. 1180 : 1  out of 20
range 1181 .. 1200 : 4  out of 20
range 1201 .. 1220 : 5  out of 20
range 1221 .. 1240 : 5  out of 20
range 1241 .. 1260 : 1  out of 20
range 1261 .. 1280 : 2  out of 20


I see that this supports my claim that it is not unusual that there
are n that fulfill the modulo relation for n near 590.
In any of the intervals checked up to index 1260, there is roughly a >=10
percent change to find such an n.  Of course you can arbitrarily reduce that
chance by i) removing odd primes, ii) removing powers of some arbitrary base
(iii) removing numbers of the form A000111(n) == 1 mod (n), but this does not
have anything to do with my observation. (It's like chopping the Amazon rain
forest and then claiming that wood is not a natural habitat there...)

Richard Mathar