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%I #17 Jul 05 2022 23:30:40
%S 0,1,0,0,2,1,1,1,0,3,0,0,0,2,2,0,1,1,2,2,1,1,0,1,0,1,0,1,4,3,0,1,0,2,
%T 3,0,0,3,0,3,1,2,3,0,2,1,0,0,0,1,1,0,1,1,2,2,2,5,0,2,1,1,1,0,2,1,2,1,
%U 0,4,0,1,3,1,0,2,1,1,1,2,0,2,0,1,3,4,4,1,0,3,1,0,0,1,4,1,0,1,0,0,2,2,1,1,3
%N Difference between the 2-adic valuation of A003973(n) [= the sum of divisors of the prime shifted n] and the 2-adic valuation of the number of divisors of n.
%C Note that A295664(n) = A295664(A003961(n)).
%H Antti Karttunen, <a href="/A336931/b336931.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F Additive with a(p^e) = 0 when e is even, a(p^e) = A007814(1+A003961(p))-1 when e is odd.
%F a(n) = A336932(n) - A295664(n).
%F a(n) = a(A007913(n)).
%o (PARI)
%o A007814(n) = valuation(n, 2);
%o A336931(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))-1)); };
%o (PARI)
%o A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
%o A007814(n) = valuation(n, 2);
%o A336931(n) = (A007814(A003973(n)) - A007814(numdiv(n)));
%o (Python)
%o from math import prod
%o from sympy import factorint, nextprime, divisor_count
%o def A336931(n): return (~(m:=prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()))& m-1).bit_length()-(~(k:=int(divisor_count(n))) & k-1).bit_length() # _Chai Wah Wu_, Jul 05 2022
%Y Cf. A003961, A003973, A007814, A007913, A295664, A336930 (positions of zeros), A336932, A336937.
%K nonn
%O 1,5
%A _Antti Karttunen_, Aug 17 2020
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