A354100 The 3-adic valuation of sigma, sum of divisors function.

0, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 0, 2, 1, 3, 1, 0, 0, 1, 0, 2, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 0, 3, 1, 2, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 0, 2, 1, 2, 2, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 2, 2, 2, 0, 1, 0, 2, 1, 2, 0, 2, 1, 0, 1, 3, 0, 1, 1

Offset: 1

Author

Antti Karttunen, May 17 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A007949, A074941.

Cf. A329963 (positions of zeros), A087943 (of terms > 0).

Cf. also A336937, A354099.

Programs

Mathematica

a[n_] := IntegerExponent[DivisorSigma[1, n], 3]; Array[a, 100] (* Amiram Eldar, May 18 2023 *)

PARI

A354100(n) = valuation(sigma(n),3);

PARI

A354100(n) = { my(f=factor(n)); sum(k=1,#f~,valuation(((f[k,1]^(f[k,2]+1))-1)/(f[k,1]-1), 3)); }; \\ Demonstrates the additivity

Formula

a(n) = A007949(A000203(n)).

Additive with a(p^e) = A007949((p^(e+1)-1)/(p-1)).