A348944 a(n) = (1/2) * (A003959(n)+A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

1, 3, 4, 7, 6, 12, 8, 18, 13, 18, 12, 28, 14, 24, 24, 49, 18, 39, 20, 42, 32, 36, 24, 72, 31, 42, 46, 56, 30, 72, 32, 138, 48, 54, 48, 97, 38, 60, 56, 108, 42, 96, 44, 84, 78, 72, 48, 196, 57, 93, 72, 98, 54, 138, 72, 144, 80, 90, 60, 168, 62, 96, 104, 397, 84, 144, 68, 126, 96, 144, 72, 261, 74, 114, 124, 140, 96

Offset: 1

Author

Antti Karttunen, Nov 05 2021

Keywords

• nonn

Comments

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 97 != 91 = 7*13 = a(4)*a(9).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Arithmetic mean of A003959 and A034448.

Cf. A348732, A348733, A348945.

Cf. also A325973.

Programs

Mathematica

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f) / 2; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

PARI

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));