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%I #20 Dec 09 2022 07:07:16
%S 0,0,0,2,0,0,0,6,3,0,0,8,0,0,0,14,0,9,0,12,0,0,0,24,5,0,12,16,0,0,0,
%T 30,0,0,0,41,0,0,0,36,0,0,0,24,18,0,0,56,7,15,0,28,0,36,0,48,0,0,0,48,
%U 0,0,24,62,0,0,0,36,0,0,0,105,0,0,20,40,0,0,0,84,39,0,0,64,0,0,0,72,0,54,0
%N Sum of non-unitary divisors of n.
%H Antti Karttunen, <a href="/A048146/b048146.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A000203(n) - A034448(n) = sigma(n) - usigma(n). a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = (p^k - p) / (p - 1), for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k >=2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * (1 - 1/zeta(3)) = 0.1382506... . - _Amiram Eldar_, Dec 09 2022
%e If n = 1000, the 12 non-unitary divisors are {2, 4, 5, 10, 20, 25, 40, 50, 100, 200, 250, 500} and their sum is a(n) = a(1000) = 1206. a(16) = a(2^4) = (2^4 - 2) / (2 - 1)= 14.
%t us[n_Integer] := (d = Divisors[n]; l = Length[d]; k = 1; s = n; While[k < l, If[ GCD[ d[[k]], n/d[[k]] ] == 1, s = s + d[[k]]]; k++ ]; s); Table[ DivisorSigma[1, n] - us[n], {n, 1, 100} ]
%t (* Second program: *)
%t Table[DivisorSum[n, # &, ! CoprimeQ[#, n/#] &], {n, 91}] (* _Michael De Vlieger_, Nov 20 2017 *)
%o (PARI) a(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1) \\ _Charles R Greathouse IV_, Jun 17 2015
%Y Cf. A034444, A000203, A006881, A048105, A048106, A048107, A048109, A048111, A005117, A120944.
%Y Cf. A002117, A072691.
%K nonn
%O 1,4
%A _Labos Elemer_
%E Edited by _Jaroslav Krizek_, Mar 01 2009
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