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A336937
The 2-adic valuation of sigma(n), the sum of divisors of n.
11
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)).
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 A069585 A199238
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 12 2020
STATUS
approved