A348049 a(n) = A003959(n) / gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

1, 1, 1, 9, 1, 1, 1, 9, 16, 1, 1, 9, 1, 1, 1, 81, 1, 16, 1, 9, 1, 1, 1, 9, 36, 1, 8, 9, 1, 1, 1, 27, 1, 1, 1, 144, 1, 1, 1, 9, 1, 1, 1, 9, 16, 1, 1, 81, 64, 36, 1, 9, 1, 8, 1, 9, 1, 1, 1, 9, 1, 1, 16, 729, 1, 1, 1, 9, 1, 1, 1, 144, 1, 1, 36, 9, 1, 1, 1, 81, 256, 1, 1, 9, 1, 1, 1, 9, 1, 16, 1, 9, 1, 1, 1, 27, 1, 64

Offset: 1

Author

Antti Karttunen, Oct 21 2021

Keywords

• nonn

Comments

Not multiplicative. For example, a(196) = 192 != a(4) * a(49).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003959, A005117 (positions of 1's), A348029, A348047, A348048.

Cf. also A344697.

Programs

Mathematica

f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := (m = Times @@ f @@@ FactorInteger[n]) / GCD[m, DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Oct 21 2021 *)

PARI

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348049(n) = { my(u=A003959(n)); (u/gcd(u, sigma(n))); };

Formula

a(n) = A003959(n) / A348047(n) = A003959(n) / gcd(A000203(n), A003959(n)).