A183098 a(1) = 0, a(n) = sum of divisors d of n such that if d = Product_{i} (p_i^e_i) then not all e_i are > 1.

0, 2, 3, 2, 5, 11, 7, 2, 3, 17, 11, 23, 13, 23, 23, 2, 17, 29, 19, 37, 31, 35, 23, 47, 5, 41, 3, 51, 29, 71, 31, 2, 47, 53, 47, 41, 37, 59, 55, 77, 41, 95, 43, 79, 68, 71, 47, 95, 7, 67, 71, 93, 53, 83, 71, 107, 79, 89, 59, 163, 61, 95, 94, 2, 83, 143, 67, 121, 95, 143, 71, 65, 73, 113, 98, 135, 95, 167, 79, 157, 3, 125, 83, 219, 107, 131, 119, 167, 89, 224, 111, 163, 127, 143, 119, 191, 97, 121, 146, 87

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn
• easy

Comments

a(n) = sum of non-powerful divisors d of n where powerful numbers are numbers from A001694(m) for m >= 2.

Sequence is not the same as A183101(n): a(72) = 65, A183101(72) = 137.

Examples

			For n = 12, the set of such divisors is {2, 3, 6, 12}; a(12) = 2+3+6+12 = 23.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16385
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000203, A001694, A183097, A183100, A183101, A183103.

Programs

Mathematica

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := f1[p, e] - p; a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

PARI

A183098(n) = sumdiv(n, d, d*(!ispowerful(d))); \\ Antti Karttunen, Oct 07 2017

Formula

a(n) = A000203(n) - A183097(n) = A183100(n) - 1.

a(1) = 0, a(p) = p, a(p*q) = p+q+p*q, a(p*q*...*z) = (p+1)*(q+1)*...*(z+1) - 1, a(p^k) = p, for p, q = primes, k = natural numbers, p*q*...*z = product of k (k > 2) distinct primes p, q, ..., z.

Extensions

Name corrected by Jon E. Schoenfield, Aug 29 2023