A086455 -id:A086455 - cp's OEIS frontend

A091051 Sum of divisors of n that are perfect powers.

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 29, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 61, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 125, 1, 1, 1, 5, 1, 1, 1, 58, 1, 1, 26, 5, 1, 1, 1, 29, 118, 1, 1, 5, 1, 1, 1, 13, 1, 10

Offset: 1

Reinhard Zumkeller, Dec 15 2003

nonn

a(n) = 1 iff n is squarefree: a(A005117(n))=1, a(A013929(n))>1;
a(p^k) = 1+(p^2)*(p^(k-1)-1)/(p-1) for p prime, k>0.
a(A000961(n)) = A086455(n)-A025473(n).

			Divisors of n=108: {1,2,3,4,6,9,12,18,27,36,54,108}, a(108) = 1^2 + 2^2 + 3^2 + 3^3 + 6^2 = 1+4+9+27+36 = 77.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Eric Weisstein's World of Mathematics, Perfect Power
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for sequences related to sums of divisors

Cf. A091050, A001597, A000203, A183104.

Differs from A183097 for the first time at n=72.

a[n_] := DivisorSum[n, #*Boole[# == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1]&]; Array[a, 90] (* Jean-François Alcover, May 09 2017 *)

a(n) = sumdiv(n, d, d*((d==1) || ispower(d))); \\ Michel Marcus, Oct 02 2014

G.f.: Sum_{k=i^j, i>=1, j>=2, excluding duplicates} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 20 2017

A086454 Number of divisors of prime powers: tau(p^e).

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 4, 3, 2, 2, 5, 2, 2, 2, 3, 4, 2, 2, 6, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 9, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jul 20 2003

nonn

Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Prime Power.

Cf. A000005, A000961, A025474, A086455.

f[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 2]] + 1, Nothing]]; f[1] = 1; Array[f, 400] (* Amiram Eldar, Apr 09 2024 *)

a(n) = A000005(A000961(n)).
a(n) = e+1 for A000961(n) = p^e.
a(n) = A025474(n) + 1.

A273938 Sum of the divisors of the n-th odd prime power.

Original entry on oeis.org

1, 4, 6, 8, 13, 12, 14, 18, 20, 24, 31, 40, 30, 32, 38, 42, 44, 48, 57, 54, 60, 62, 68, 72, 74, 80, 121, 84, 90, 98, 102, 104, 108, 110, 114, 133, 156, 128, 132, 138, 140, 150, 152, 158, 164, 168, 183, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230

Offset: 1

R. J. Mathar, Jun 04 2016

nonn

Cf. A061345, A086455, A247837.

A273938 := proc(n)
        numtheory[sigma](A061345(n-1)) ;
end proc:

a(n) = A000203(A061345(n-1)).

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A091051 Sum of divisors of n that are perfect powers.

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A086454 Number of divisors of prime powers: tau(p^e).

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A273938 Sum of the divisors of the n-th odd prime power.

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