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The 3-adic valuation of sigma, sum of divisors function.
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%I #15 May 18 2023 10:40:39

%S 0,1,0,0,1,1,0,1,0,2,1,0,0,1,1,0,2,1,0,1,0,2,1,1,0,1,0,0,1,2,0,2,1,3,

%T 1,0,0,1,0,2,1,1,0,1,1,2,1,0,1,1,2,0,3,1,2,1,0,2,1,1,0,1,0,0,1,2,0,2,

%U 1,2,2,1,0,1,0,0,1,1,0,1,0,2,1,0,3,1,1,2,2,2,0,1,0,2,1,2,0,2,1,0,1,3,0,1,1

%N The 3-adic valuation of sigma, sum of divisors function.

%H Antti Karttunen, <a href="/A354100/b354100.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) = A007949(A000203(n)).

%F Additive with a(p^e) = A007949((p^(e+1)-1)/(p-1)).

%t a[n_] := IntegerExponent[DivisorSigma[1, n], 3]; Array[a, 100] (* _Amiram Eldar_, May 18 2023 *)

%o (PARI) A354100(n) = valuation(sigma(n),3);

%o (PARI) A354100(n) = { my(f=factor(n)); sum(k=1,#f~,valuation(((f[k,1]^(f[k,2]+1))-1)/(f[k,1]-1), 3)); }; \\ Demonstrates the additivity

%Y Cf. A000203, A007949, A074941.

%Y Cf. A329963 (positions of zeros), A087943 (of terms > 0).

%Y Cf. also A336937, A354099.

%K nonn

%O 1,10

%A _Antti Karttunen_, May 17 2022