A336937 The 2-adic valuation of sigma(n), the sum of divisors of n.

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

Offset: 1

Author

Antti Karttunen, Aug 12 2020

Keywords

• nonn

Comments

Equally, the 2-adic valuation of A000593(n), the sum of odd divisors of n.

Proof for the given additive formula: It's easy to see that for all powers of 2 and all even powers of odd primes the result is zero. Thus assuming p is an odd prime, factorize sigma(p^(2e-1)) = (1 + p + p^2 + ... + p^(2e-1)) as (1+p)*(1 + u + u^2 + u^3 + ... + u^(e-1)), where u=p^2. Note that u [and its powers] are always of the form 4k+1, thus the 2-adic valuation of that sum is A007814(e) [see my Aug 15 2020 comment there] which when added to the 2-adic valuation of 1+p then gives the 2-adic valuation for whole sigma(p^(2e-1)).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to sigma(n)

Crossrefs

One less than A286357.

Cf. A000035, A000203, A000265, A000593, A007814, A028982 (positions of zeros), A161942, A295664, A336932.

Programs

Mathematica

a[n_] := IntegerExponent[DivisorSigma[1, n], 2]; Array[a, 100] (* Amiram Eldar, Jul 04 2022 *)

PARI

A336937(n) = valuation(sigma(n),2);

PARI

A007814(n) = valuation(n,2);
A336937(n) = { my(f=factor(n)); sum(i=1,#f~,(f[i,1]%2) * (f[i,2]%2) * (A007814(1+f[i,1])+A007814(1+f[i,2])-1)); };

Python

from sympy import divisor_sigma
def A336937(n): return (~(m:=int(divisor_sigma(n))) & m-1).bit_length() # Chai Wah Wu, Jul 01 2022

Formula

a(n) = A007814(A000203(n)) = A007814(A000593(n)).

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

For n with factorization p_1^e_1 * ... * p_k^e_k, a(n) = Sum_{i=1..k} A000035(p_i) * A000035(e_i) * (A007814(1+p_i)+A007814(1+e_i)-1).

a(n) = A286357(n)-1.

a(n) = a(2*n) = a(A000265(n)).

For any odd n, a(n) >= A295664(n).

a(A003961(n)) = A336932(n).

A161942(n) = A000203(n) / (2^a(n)).