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

1,3

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)

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)).

Mathematica

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

Prog

(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

Crossrefs

One less than A286357.

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

Sequence in context: A080099 A268040 A127711 * A069584 A199238 A346698

Adjacent sequences: A336934 A336935 A336936 * A336938 A336939 A336940

Keyword

nonn

Author

Antti Karttunen, Aug 12 2020

Status

approved