A349756 Numbers k such that the odd part of sigma(k) is equal to gcd(sigma(k), A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

1, 2, 3, 6, 7, 14, 20, 21, 24, 27, 31, 42, 54, 57, 60, 62, 93, 114, 120, 127, 140, 160, 168, 186, 189, 216, 217, 220, 237, 254, 264, 301, 378, 381, 399, 408, 420, 434, 460, 474, 480, 513, 540, 552, 602, 620, 651, 660, 744, 762, 792, 798, 837, 840, 889, 903, 940, 1026, 1080, 1120, 1128, 1140, 1302, 1320, 1380, 1392, 1512

Offset: 1

Author

Antti Karttunen, Dec 03 2021

Keywords

• nonn

Comments

Numbers k for which A161942(k) = A342671(k).

From Antti Karttunen, Jul 23 2022: (Start)

Numbers k such that k is a multiple of A350073(k).

For any square s in this sequence, A349162(s) = 1, i.e. sigma(s) divides A003961(s), and also A286385(s). Question: Is 1 the only square in this sequence? (see the conjecture in A350072).

If both x and y are terms and gcd(x, y) = 1, then x*y is also present.

After 2, the only primes present are Mersenne primes, A000668.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..23008; terms < 2^31
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Positions of 1's in A348992.

Positions where the powers of 2 (A000079) occur in A349162.

Cf. A000203, A003961, A161942, A286385, A342671, A350072, A350073, A355946 (characteristic function).

Cf. A000668, A046528 (subsequences).

Cf. also A348943.

Programs

Mathematica

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; oddpart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := oddpart[(sigma = DivisorSigma[1, n])] == GCD[sigma, s[n]]; Select[Range[1500], q] (* Amiram Eldar, Dec 04 2021 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A355946(n) = { my(s=sigma(n)); !(A003961(n)%((s>>=valuation(s,2)))); };
isA349756(n) = A355946(n);