A319687 - OEIS

%I #16 May 19 2023 01:50:06

%S 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,

%T 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,

%U 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

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

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

%H Antti Karttunen, <a href="/A319687/b319687.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>

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

%o (PARI)

%o A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487

%o A318509(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(f[i, 1])); factorback(f); };

%o A319687(n) = (A318509(n) - A002487(n));

%o (Python)

%o from math import prod

%o from functools import reduce

%o from sympy import factorint

%o 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

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

%K sign,look

%O 1,15

%A _Antti Karttunen_, Oct 02 2018