A334863 a(n) = A064097(A003961(n)).

0, 2, 3, 4, 4, 5, 5, 6, 6, 6, 5, 7, 5, 7, 7, 8, 6, 8, 7, 8, 8, 7, 7, 9, 8, 7, 9, 9, 7, 9, 7, 10, 8, 8, 9, 10, 7, 9, 8, 10, 8, 10, 9, 9, 10, 9, 8, 11, 10, 10, 9, 9, 9, 11, 9, 11, 10, 9, 8, 11, 9, 9, 11, 12, 9, 10, 9, 10, 10, 11, 8, 12, 9, 9, 11, 11, 10, 10, 9, 12, 12, 10, 9, 12, 10, 11, 10, 11, 8, 12, 10, 11, 10, 10

Offset: 1

Author

Antti Karttunen, May 19 2020

Keywords

• nonn
• less

Comments

Completely additive because A003961 is fully multiplicative and A064097 is fully additive.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003961, A064097, A334864.

Programs

PARI

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f);
A064097(n) = if(1==n,0,1+A064097(n-(n/vecmin(factor(n)[,1]))));
A334863(n) = A064097(A003961(n));
\\ Or alternatively as:
A334863(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*A064097(prime(1+primepi(f[k, 1])))); };

Formula

a(n) = A064097(A003961(n)).

a(1) = 0; and for n > 1, a(prime(k)) = A064097(prime(1+k)) for k-th prime, and a(n*m) = a(n) + a(m) if m,n > 1.