cp's OEIS Frontend

A316907 Numbers k such that 2^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.

%I A316907 #50 Oct 28 2021 06:36:29
%S A316907 7957,23377,35333,42799,49981,60787,129889,150851,162193,164737,
%T A316907 241001,249841,253241,256999,280601,318361,452051,481573,556169,
%U A316907 580337,617093,665333,722201,838861,877099,1016801,1251949,1252697,1325843,1507963,1534541,1678541,1826203,1969417
%N A316907 Numbers k such that 2^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.
%C A316907 Numbers k such that 2^k == 2 (mod k) and gcd(k,b^k-b) = 1 for some b > 2.
%C A316907 Problem: are there infinitely many such "anti-Carmichael pseudoprimes"?
%C A316907 All semiprime terms of A316906 are in the sequence; i.e., numbers m in A214305 such that p-1 does not divide q-1, where m = pq and p < q are primes.
%H A316907 Amiram Eldar, <a href="/A316907/b316907.txt">Table of n, a(n) for n = 1..10000</a>
%e A316907 7957 = 73*109 is pseudoprime and 72 does not divide 7956 (of course also 108 does not divide 7956), note that 72 does not divide 108.
%e A316907 617093 = 43*113*127 is the smallest such pseudoprime with more than two prime factors.
%t A316907 Select[Select[Range[2*10^6], PowerMod[2, # - 1, #] == 1 &], Function[k, AllTrue[FactorInteger[k][[All, 1]] - 1, Mod[k - 1, #] != 0 &]]] (* _Michael De Vlieger_, Jul 20 2018 *)
%Y A316907 Subsequence of A001567 and of A316906.
%Y A316907 Cf. A121707 (probably "anti-Carmichael numbers": n such that p-1 does not divide n-1 for every prime p dividing n).
%K A316907 nonn
%O A316907 1,1
%A A316907 _Thomas Ordowski_, Jul 16 2018
%E A316907 a(7)-a(34) from _Michel Marcus_, Jul 16 2018