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.

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, 4, 52, 30, 72, 32, 3, 48, 54, 48, 42, 38, 60, 56, 78, 42, 96, 44, 80, 69, 72, 48, 96, 8, 68, 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

Offset

1,2

Comments

a(n) = sum of non-powerful divisors d of n where powerful numbers are numbers from A001694(m) for m >= 1.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16385

Index entries for sequences related to sums of divisors.

Formula

a(n) = A000203(n) - A183099(n) = A183098(n) + 1.

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.

Example

For n = 12, the set of such divisors is {1, 2, 3, 6, 12}; a(12) = 1+2+3+6+12 = 24.

Mathematica

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 *)

Prog

(PARI) A183100(n) = (1 + sumdiv(n, d, d*(!ispowerful(d)))); \\ Antti Karttunen, Oct 07 2017

Crossrefs

Cf. A000203, A001694, A183098, A183099.

Sequence in context: A048250 A323363 A073181 * A340323 A046897 A109506

Adjacent sequences: A183097 A183098 A183099 * A183101 A183102 A183103

Keyword

nonn,easy

Author

Jaroslav Krizek, Dec 25 2010

Status

approved