A318510 Completely multiplicative with a(prime(k)) = A002487(prime(k+1)).

1, 2, 3, 4, 3, 6, 5, 8, 9, 6, 5, 12, 5, 10, 9, 16, 7, 18, 7, 12, 15, 10, 7, 24, 9, 10, 27, 20, 5, 18, 11, 32, 15, 14, 15, 36, 11, 14, 15, 24, 13, 30, 9, 20, 27, 14, 13, 48, 25, 18, 21, 20, 11, 54, 15, 40, 21, 10, 9, 36, 11, 22, 45, 64, 15, 30, 13, 28, 21, 30, 15, 72, 13, 22, 27, 28, 25, 30, 19, 48, 81, 26, 17, 60, 21, 18, 15, 40

Offset: 1

Author

Antti Karttunen, Aug 30 2018

Keywords

• nonn
• mult
• look

Comments

Provided that the conjecture given in A261179 holds, then for all n >= 1, A007814(a(n)) = A007814(n), i.e., then the sequence preserves the 2-adic valuation of 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, A003961, A318509.

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
A318510(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(prime(1+primepi(f[i, 1])))); factorback(f); };

Python

from math import prod
from functools import reduce
from sympy import factorint, nextprime
def A318510(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(nextprime(p))[-1:2:-1],(1,0)))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 18 2023

Formula

a(n) = A318509(A003961(n)).