This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336937 #52 Jul 04 2022 04:38:45
%S A336937 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,
%T A336937 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,
%U A336937 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
%N A336937 The 2-adic valuation of sigma(n), the sum of divisors of n.
%C A336937 Equally, the 2-adic valuation of A000593(n), the sum of odd divisors of n.
%C A336937 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)).
%H A336937 Antti Karttunen, <a href="/A336937/b336937.txt">Table of n, a(n) for n = 1..65537</a>
%H A336937 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A336937 a(n) = A007814(A000203(n)) = A007814(A000593(n)).
%F A336937 Additive with a(2^e) = 0, a(p^2e) = 0, a(p^(2e-1)) = A007814(1+p) + A007814(e).
%F A336937 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).
%F A336937 a(n) = A286357(n)-1.
%F A336937 a(n) = a(2*n) = a(A000265(n)).
%F A336937 For any odd n, a(n) >= A295664(n).
%F A336937 a(A003961(n)) = A336932(n).
%F A336937 A161942(n) = A000203(n) / (2^a(n)).
%t A336937 a[n_] := IntegerExponent[DivisorSigma[1, n], 2]; Array[a, 100] (* _Amiram Eldar_, Jul 04 2022 *)
%o A336937 (PARI) A336937(n) = valuation(sigma(n),2);
%o A336937 (PARI)
%o A336937 A007814(n) = valuation(n,2);
%o A336937 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)); };
%o A336937 (Python)
%o A336937 from sympy import divisor_sigma
%o A336937 def A336937(n): return (~(m:=int(divisor_sigma(n))) & m-1).bit_length() # _Chai Wah Wu_, Jul 01 2022
%Y A336937 One less than A286357.
%Y A336937 Cf. A000035, A000203, A000265, A000593, A007814, A028982 (positions of zeros), A161942, A295664, A336932.
%K A336937 nonn
%O A336937 1,3
%A A336937 _Antti Karttunen_, Aug 12 2020