A318509 -id:A318509 - cp's OEIS frontend

A319687 a(n) = A318509(n) - A002487(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 4, 0, 4, 0, 0, 2, 0, 0, 6, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, -2, -6, 0, -4, 0, 0, 0, -4, 0, -6, 0, 10, 0, 0, 0, 4, 2, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 02 2018

All terms seem to be even. See the conjecture given in A261179.

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

Cf. A002487, A261179, A317837, A318509, A323365.

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); };
A319687(n) = (A318509(n) - A002487(n));

from math import prod
from functools import reduce
from sympy import factorint
def A319687(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())-sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) # Chai Wah Wu, May 18 2023

a(n) = A318509(n) - A002487(n).

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

Original entry on oeis.org

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

Antti Karttunen, Aug 30 2018

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.

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

Cf. A002487, A003961, A318509.

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); };

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

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

cp's OEIS Frontend

A319687 a(n) = A318509(n) - A002487(n).

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