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.

8911, 29341, 314821, 410041, 1024651, 1152271, 5481451, 10267951, 14913991, 15247621, 36765901, 64377991, 67902031, 133800661, 139952671, 178482151, 188516329, 299736181, 362569201, 368113411, 395044651, 532758241, 579606301, 612816751, 620169409, 625482001, 652969351

Offset: 1

Author

Amiram Eldar, Apr 05 2025

Keywords

• nonn

Comments

Subsequence of A329799 and first differs from it at n = 36: A329799(36) = 1419339691 is not a term of this sequence.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..12071 (terms below 2^64)
• R. Balasubramanian and S. V. Nagaraj, The least witness of a composite number, International Workshop on Information Security, Springer, Berlin, Heidelberg, 1997, pp. 66-74.
• S. V. Nagaraj, Problems in Algorithmic Number theory, Ph.D. thesis, University of Madras, 1999. See Chapter 5, section 5.3, p. 43.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Subsequence of A002997, A087788 and A329799.

Cf. A007814.

Programs

Mathematica

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1} && SameQ @@ IntegerExponent[f[[;; , 1]] - 1, 2]];
Select[Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]], q]

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);