A336932 The 2-adic valuation of A003973(n), the sum of divisors of prime shifted n.

0, 2, 1, 0, 3, 3, 2, 3, 0, 5, 1, 1, 1, 4, 4, 0, 2, 2, 3, 3, 3, 3, 1, 4, 0, 3, 2, 2, 5, 6, 1, 2, 2, 4, 5, 0, 1, 5, 2, 6, 2, 5, 4, 1, 3, 3, 1, 1, 0, 2, 3, 1, 2, 4, 4, 5, 4, 7, 1, 4, 2, 3, 2, 0, 4, 4, 3, 2, 2, 7, 1, 3, 4, 3, 1, 3, 3, 4, 2, 3, 0, 4, 1, 3, 5, 6, 6, 4, 1, 5, 3, 1, 2, 3, 6, 3, 1, 2, 1, 0, 3, 5, 2, 4, 6

Offset

1,2

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)

Formula

Additive with a(p^2e) = 0, a(p^(2e-1)) = A007814(1+A003961(p)) + A007814(e).

a(n) = A007814(A003973(n)).

a(n) = A336937(A003961(n)).

For all n >= 1, a(n) >= A295664(n).

Prog

(PARI) A336932(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); valuation(sigma(factorback(f)), 2); };
(PARI)
A007814(n) = valuation(n, 2);
A336932(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))+A007814(1+f[i, 2])-1)); };
(Python)
from math import prod
from sympy import factorint, nextprime
def A336932(n): return (~(m:=prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p, e in factorint(n).items()))& m-1).bit_length() # Chai Wah Wu, Jul 05 2022

Crossrefs

Cf. A000203, A000290 (positions of zeros), A003961, A003973, A007814, A295664, A336937.

Sequence in context: A347277 A357734 A228821 * A127258 A154557 A049242

Adjacent sequences: A336929 A336930 A336931 * A336933 A336934 A336935

Keyword

nonn

Author

Antti Karttunen, Aug 16 2020

Status

approved