A326041 a(n) = sigma(A064989(n)).

1, 1, 3, 1, 4, 3, 6, 1, 7, 4, 8, 3, 12, 6, 12, 1, 14, 7, 18, 4, 18, 8, 20, 3, 13, 12, 15, 6, 24, 12, 30, 1, 24, 14, 24, 7, 32, 18, 36, 4, 38, 18, 42, 8, 28, 20, 44, 3, 31, 13, 42, 12, 48, 15, 32, 6, 54, 24, 54, 12, 60, 30, 42, 1, 48, 24, 62, 14, 60, 24, 68, 7, 72, 32, 39, 18, 48, 36, 74, 4, 31, 38, 80, 18, 56, 42, 72, 8

Offset: 1

Author

Antti Karttunen, Jun 03 2019

Keywords

• nonn
• mult

Comments

The odd bisection is A285703, the even bisection is the sequence itself.

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, A003973, A064989, A151799, A285703.

Programs

PARI

A326041(n) = if(1==n,n, my(f = factor(n)); prod(i=1, #f~, if(2==f[i,1],1,((precprime(f[i,1]-1)^(1+f[i,2]))-1)/(precprime(f[i,1]-1)-1))));

PARI

A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
A326041(n) = sigma(A064989(n));

Formula

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (q^(e+1)-1)/(q-1), where q = A151799(p).

a(n) = A000203(A064989(n)).

a(2n) = a(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/3) * Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.2722825585..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023