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A382791 Carmichael numbers with exactly 3 prime factors, p*q*r, such that p-1, q-1 and r-1 have an equal 2-adic valuation.

%I A382791 #9 Apr 05 2025 09:10:49
%S A382791 8911,29341,314821,410041,1024651,1152271,5481451,10267951,14913991,
%T A382791 15247621,36765901,64377991,67902031,133800661,139952671,178482151,
%U A382791 188516329,299736181,362569201,368113411,395044651,532758241,579606301,612816751,620169409,625482001,652969351
%N A382791 Carmichael numbers with exactly 3 prime factors, p*q*r, such that p-1, q-1 and r-1 have an equal 2-adic valuation.
%C A382791 Subsequence of A329799 and first differs from it at n = 36: A329799(36) = 1419339691 is not a term of this sequence.
%H A382791 Amiram Eldar, <a href="/A382791/b382791.txt">Table of n, a(n) for n = 1..12071</a> (terms below 2^64)
%H A382791 R. Balasubramanian and S. V. Nagaraj, <a href="https://doi.org/10.1007/BFb0030409">The least witness of a composite number</a>, International Workshop on Information Security, Springer, Berlin, Heidelberg, 1997, pp. 66-74.
%H A382791 S. V. Nagaraj, <a href="https://www.imsc.res.in/xmlui/handle/123456789/85">Problems in Algorithmic Number theory</a>, Ph.D. thesis, University of Madras, 1999. See Chapter 5, section 5.3, p. 43.
%H A382791 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%e A382791 8911 = 7 * 19 * 67 is a term since it is a Carmichael number, it has exactly 3 prime factors, and 7 - 1 = 2*3, 19 - 1 = 2*3^2, and 67 - 1 = 2*3*11 all have 2-adic valuation 1.
%t A382791 q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1} && SameQ @@ IntegerExponent[f[[;; , 1]] - 1, 2]];
%t A382791 Select[Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {_, _}][[;; , 2]], q]
%o A382791 (PARI) isok(k) = if(!(k % 2) || isprime(k), 0, my(f = factor(k)); #f~ == 3 && k % lcm(znstar(k)[2]) == 1 && #Set(apply(x -> valuation(x-1, 2), f[,1])) == 1);
%Y A382791 Subsequence of A002997, A087788 and A329799.
%Y A382791 Cf. A007814.
%K A382791 nonn
%O A382791 1,1
%A A382791 _Amiram Eldar_, Apr 05 2025