A318509 Completely multiplicative with a(p) = A002487(p).

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 6, 1, 5, 4, 7, 3, 6, 5, 7, 2, 9, 5, 8, 3, 7, 6, 5, 1, 10, 5, 9, 4, 11, 7, 10, 3, 11, 6, 13, 5, 12, 7, 9, 2, 9, 9, 10, 5, 13, 8, 15, 3, 14, 7, 11, 6, 9, 5, 12, 1, 15, 10, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 15, 10, 13, 3, 16, 11, 19, 6, 15, 13, 14, 5, 17, 12, 15, 7, 10, 9, 21, 2, 11, 9, 20, 9, 19, 10, 17, 5, 18

Offset: 1

Author

Antti Karttunen, Aug 30 2018

Keywords

• nonn
• mult

Comments

Provided that the conjecture given in A261179 holds, then for all n >= 1, A007814(a(n)) = A007949(n).

Links

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

Crossrefs

Cf. A002487, A261179, A318510.

Cf. also A318307.

Programs

PARI

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318509(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(f[i, 1])); factorback(f); };

Python

from math import prod
from functools import reduce
from sympy import factorint
def A318509(n): return prod(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(p)[-1:2:-1],(1,0)))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 18 2023