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a(n) = A007814(sigma(n)) - A007814(n), where A007814(n) gives the 2-adic valuation of n, and sigma is the sum of divisors function.
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%I #18 Nov 23 2023 15:12:13

%S 0,-1,2,-2,1,1,3,-3,0,0,2,0,1,2,3,-4,1,-1,2,-1,5,1,3,-1,0,0,3,1,1,2,5,

%T -5,4,0,4,-2,1,1,3,-2,1,4,2,0,1,2,4,-2,0,-1,3,-1,1,2,3,0,4,0,2,1,1,4,

%U 3,-6,2,3,2,-1,5,3,3,-3,1,0,2,0,5,2,4,-3,0,0,2,3,2,1,3,-1,1,0,4,1,7,3,3,-3,1,-1,2,-2,1,2,3,-2,6

%N a(n) = A007814(sigma(n)) - A007814(n), where A007814(n) gives the 2-adic valuation of n, and sigma is the sum of divisors function.

%H Antti Karttunen, <a href="/A361025/b361025.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A336937(n) - A007814(n).

%F Additive with a(2^e) = -e, and for odd primes p, a(p^2e) = 0, a(p^(2e-1)) = A007814(1+p) + A007814(e). - _Antti Karttunen_, Nov 23 2023

%t a[n_] := IntegerExponent[DivisorSigma[1, n], 2] - IntegerExponent[n, 2]; Array[a, 100] (* _Amiram Eldar_, Mar 03 2023 *)

%o (PARI) A361025(n) = (valuation(sigma(n), 2) - valuation(n, 2));

%Y Cf. A000203, A007814, A336937.

%Y Cf. A216780 (positions of 0's), A361024 (their characteristic function), A216781 (positions of terms < 0), A216782 (positions of terms > 0), A324903 (their characteristic function).

%K sign

%O 1,3

%A _Antti Karttunen_, Mar 03 2023