A392921 Numbers k such that the odd part of psi(k) divides k-1, where psi = A002322 and odd part = A000265.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 19, 20, 23, 24, 28, 29, 30, 31, 32, 34, 37, 40, 41, 43, 47, 48, 51, 52, 53, 59, 60, 61, 64, 66, 67, 68, 70, 71, 73, 79, 80, 83, 85, 89, 91, 96, 97, 101, 102, 103, 107, 109, 112, 113, 120, 127, 128, 130, 131, 136, 137, 139, 149

Offset

1,2

Comments

Odd k is a term if and only if k is squarefree and odd(p-1) divides k-1 for every prime factor p of k.

Different from A339879 by having extra terms 364, 532, 679, 703, ...

For odd k>1, A393131((k-1)/2), the index of witnesses for Solovay-Strassen primality test of k as a subgroup of (Z/kZ)*, is a power of 2 if and only if k is in this sequence. In other words, we have odd(A329726((k-1)/2)) <= odd(EulerPhi(k)) for every odd k, where equality holds if and only if k is in this sequence.

Links

Jianing Song, Table of n, a(n) for n = 1..20000

Mathematica

q[k_] := Divisible[k-1, (# / 2^IntegerExponent[#, 2])& @ CarmichaelLambda[k]]; Select[Range[150], q] (* Amiram Eldar, Mar 30 2026 *)

Prog

(PARI) odd(n) = n >> valuation(n, 2)
isA392921(n) = ((n-1) % odd(lcm(znstar(n)[2])) == 0)

Crossrefs

Cf. A329726, A393131, A392922, A000265, A002322, A339879.

Sequence in context: A193159 A308018 A242455 * A339879 A274374 A007298

Adjacent sequences: A392918 A392919 A392920 * A392922 A392923 A392924

Keyword

nonn

Author

Jianing Song, Mar 16 2026

Status

approved