A336931 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.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Aug 17 2020

Keywords

• nonn

Comments

Note that A295664(n) = A295664(A003961(n)).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A003961, A003973, A007814, A007913, A295664, A336930 (positions of zeros), A336932, A336937.

Programs

PARI

A007814(n) = valuation(n, 2);
A336931(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))-1)); };

PARI

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A007814(n) = valuation(n, 2);
A336931(n) = (A007814(A003973(n)) - A007814(numdiv(n)));

Python

from math import prod
from sympy import factorint, nextprime, divisor_count
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

Formula

Additive with a(p^e) = 0 when e is even, a(p^e) = A007814(1+A003961(p))-1 when e is odd.

a(n) = A336932(n) - A295664(n).

a(n) = a(A007913(n)).