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%I #33 Aug 26 2019 05:05:48
%S 231,561,1045,1105,1653,1729,2465,2821,3059,3655,4371,4641,5005,5083,
%T 5365,5565,6545,6601,7337,8029,8695,8911,9361,10011,10585,10857,10879,
%U 11077,11305,12121,13685,13695,15181,15753,15841,16269,17755,18361,18565,19721,20301,20501,21115
%N Squarefree composite numbers n such that b^n == b (mod gpf(n)) for every integer b, where gpf(n) = A006530(n).
%C It suffices to check all bases 2 <= b <= (gpf(n)+1)/2.
%C Squarefree composite numbers n such that gpf(n)-1 divides n-1 (by analogy to the Korselt's criterion).
%C Gpf(n) is odd, so gpf(n)-1 is even. Therefore, n-1 is even so n is odd. - _David A. Corneth_, Sep 20 2016
%C These numbers have at least three prime factors. Carmichael numbers A002997 are a subsequence. So the sequence is infinite. - _Thomas Ordowski_ and _Altug Alkan_, Sep 20 2016
%C First numbers with 3, 4, ... prime factors are 231 = 3 * 7 * 11, 4641 = 3 * 7 * 13 * 17, 31395 = 3 * 5 * 7 * 13 * 23, 1163085 = 3 * 5 * 7 * 11 * 19 * 53, 11996985 = 3 * 5 * 7 * 11 * 13 * 17 * 47, and 286140855 = 3 * 5 * 7 * 11 * 13 * 17 * 19 * 59. - _Charles R Greathouse IV_, Sep 20 2016
%H Amiram Eldar, <a href="/A276832/b276832.txt">Table of n, a(n) for n = 1..10000</a>
%e 231 = 3*7*11 is a term because b^231 == b (mod 11) for every integer b.
%e 45 = 3^2*5 is not a term because 45 is not squarefree although b^45 == b (mod 5) for every integer b.
%t Select[Select[DeleteCases[Range[2, 22000], p_ /; PrimeQ@ p], SquareFreeQ], Divisible[# - 1, FactorInteger[#][[-1, 1]] - 1] &] (* _Michael De Vlieger_, Sep 20 2016 *)
%o (PARI) A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
%o is(n) = n>1 && issquarefree(n) && (n-1) % (A006530(n)-1) == 0 && !isprime(n)
%Y Cf. A002997, A006530, A276818.
%K nonn
%O 1,1
%A _Thomas Ordowski_ and _Altug Alkan_, Sep 20 2016
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