A350071 a(n) = gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 7, 1, 1, 13, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 1, 3, 1, 1, 7, 3, 1, 1, 1, 1, 1, 3, 121, 1, 1, 13, 1, 1, 1, 21, 1, 1, 1, 3, 1, 3, 1, 3, 1, 7, 1, 1, 1, 3, 1, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Author

Antti Karttunen, Dec 12 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A000290, A003961, A065764, A350072.

Programs

Mathematica

f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^(2*e); a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Dec 12 2021 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
A350071(n) = A342671(n^2);

Formula

a(n) = A342671(n^2).

a(n) = A065764(n) / A350072(n).