A349746 Numbers k for which k * gcd(sigma(k), u) is equal to sigma(k) * gcd(k, u), where u is obtained by shifting the prime factorization of k two steps toward larger primes [with u = A003961(A003961(k))].

1, 11466, 114660, 411264, 804384, 871416, 4999680, 46332000, 176417280, 378069120, 396168192, 485188704, 709430400, 2004912000, 3921372000, 5600534400, 6128179200, 6956471808, 7556976000, 7746979968, 9904204800, 14092001280, 14182439040, 23423662080, 31998395520

Offset: 1

Author

Antti Karttunen, Nov 30 2021

Keywords

• nonn

Comments

Sigma preserves both the 2-adic and 3-adic valuation of the terms of this sequence.

All 65 known 5-multiperfect numbers (A046060) are included in this sequence, as well as the smallest 7-multiperfect number, 141310897947438348259849402738485523264343544818565120000 = A007539(7), and probably the majority of other p-multiperfect numbers as well, where p is a prime > 3. However, any term that is in A349747 is not included in this sequence.

Links

• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A007539, A007814, A007949, A046060, A349747.

Cf. also A349169, A349745.

Programs

Mathematica

f[p_, e_] := NextPrime[p, 2]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := n * GCD[(sigma = DivisorSigma[1, n]), (u = s[n])] == sigma * GCD[n, u]; Select[Range[10^6], q] (* Amiram Eldar, Dec 01 2021 *)

PARI

A003961twice(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(1+nextprime(1+f[i, 1]))); factorback(f); };
isA349746(n) = { my(s=sigma(n),u=A003961twice(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

Formula

For all n >= 1, A007814(A000203(a(n))) = A007814(a(n)) and A007949(A000203(a(n))) = A007949(a(n)). [See comment]

Extensions

a(15)-a(25) from Martin Ehrenstein, Dec 17 2021