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A183100 a(n) is the sum of divisors d of n which are either 1 or of the form Product_{i} (p_i^e_i) where at least one e_i = 1.

%I A183100 #31 Sep 03 2023 10:43:27
%S A183100 1,3,4,3,6,12,8,3,4,18,12,24,14,24,24,3,18,30,20,38,32,36,24,48,6,42,
%T A183100 4,52,30,72,32,3,48,54,48,42,38,60,56,78,42,96,44,80,69,72,48,96,8,68,
%U A183100 72,94,54,84,72,108,80,90,60,164,62,96,95,3,84,144,68,122,96,144,72,66,74,114,99,136,96,168,80,158,4,126,84,220,108,132,120,168,90,225,112,164,128,144,120,192,98,122,147,88
%N A183100 a(n) is the sum of divisors d of n which are either 1 or of the form Product_{i} (p_i^e_i) where at least one e_i = 1.
%C A183100 a(n) = sum of non-powerful divisors d of n where powerful numbers are numbers from A001694(m) for m >= 1.
%H A183100 Antti Karttunen, <a href="/A183100/b183100.txt">Table of n, a(n) for n = 1..16385</a>
%H A183100 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A183100 a(n) = A000203(n) - A183099(n) = A183098(n) + 1.
%F A183100 a(1) = 1, a(p) = p+1, a(p*q) = (p+1)*(q+1), a(p*q*...*z) = (p+1)*(q+1)*...*(z+1), a(p^k) = p+1, for p, q = primes, k = natural numbers, p*q*...*z = product of k (k > 2) distinct primes p, q, ..., z.
%e A183100 For n = 12, the set of such divisors is {1, 2, 3, 6, 12}; a(12) = 1+2+3+6+12 = 24.
%t A183100 f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := f1[p, e] - p; a[1] = 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f + 1; Array[a, 100] (* _Amiram Eldar_, Aug 29 2023 *)
%o A183100 (PARI) A183100(n) = (1 + sumdiv(n, d, d*(!ispowerful(d)))); \\ _Antti Karttunen_, Oct 07 2017
%Y A183100 Cf. A000203, A001694, A183098, A183099.
%K A183100 nonn,easy
%O A183100 1,2
%A A183100 _Jaroslav Krizek_, Dec 25 2010