A183104 -id:A183104 - 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

A183102 a(n) = product of powerful divisors d of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 32, 9, 1, 1, 4, 1, 1, 1, 512, 1, 9, 1, 4, 1, 1, 1, 32, 25, 1, 243, 4, 1, 1, 1, 16384, 1, 1, 1, 1296, 1, 1, 1, 32, 1, 1, 1, 4, 9, 1, 1, 512, 49, 25, 1, 4, 1, 243, 1, 32, 1, 1, 1, 4, 1, 1, 9, 1048576, 1, 1, 1, 4, 1, 1, 1, 746496, 1

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

a(n) = product of divisors d of n from set A001694 - powerful numbers.
Sequence is not the same as A183104(n): a(72) = 746496, A183104(72) = 10368.
Not multiplicative: a(4)*a(9) = 4*9=36 <> a(36) = 1296. - R. J. Mathar, Jun 07 2011

			For n = 12, set of such divisors is {1, 4}; a(12) = 1*4 = 4.

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

Cf. A001694, A007955, A183097, A183103, A183104.

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A183102 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if isA001694(d) then a := a*d; end if; end do; a ; end proc:
seq(A183102(n),n=1..70) ; # R. J. Mathar, Jun 07 2011

powerfulQ[n_] := Min[FactorInteger[n][[All, 2]]] > 1;
a[n_] := Times @@ Select[Divisors[n], powerfulQ];
Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Jun 01 2024 *)

A183102(n) = { my(m=1); fordiv(n, d, if(ispowerful(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

a(n) = A007955(n) / A183103(n).

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

A183105 a(n) = product of divisors of n that are not perfect powers.

Original entry on oeis.org

1, 2, 3, 2, 5, 36, 7, 2, 3, 100, 11, 432, 13, 196, 225, 2, 17, 648, 19, 2000, 441, 484, 23, 10368, 5, 676, 3, 5488, 29, 810000, 31, 2, 1089, 1156, 1225, 7776, 37, 1444, 1521, 80000, 41, 3111696, 43, 21296, 10125, 2116, 47, 497664, 7, 5000, 2601, 35152, 53, 34992, 3025, 307328, 3249, 3364, 59, 11664000000, 61, 3844, 27783, 2, 4225, 18974736, 67, 78608, 4761, 24010000, 71, 13436928

Offset: 1

Jaroslav Krizek, Dec 25 2010

Sequence is not the same as A183103: a(72) = 13436928, A183103(72) = 186624.

Not multiplicative since a(2)*a(3) <> a(6). - R. J. Mathar, Jun 07 2011

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 1*2*3*6*12 = 432.

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

Cf. A001597, A007955, A183101, A183103, A183104.

isA001597 := proc(n) local e ; e := seq(op(2,p),p=ifactors(n)[2]) ; return ( igcd(e) >=2 ) ; end proc:
A183105 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if not isA001597(d) then a := a*d; end if; end do; a ; end proc:
seq(A183105(n),n=1..72) ; # R. J. Mathar, Jun 07 2011

perfectPowerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1;
a[n_] := Times @@ Select[Divisors[n], !perfectPowerQ[#]&];
Table[a[n], {n, 1, 72}] (* Jean-François Alcover, May 31 2024 *)

A183105(n) = { my(m=1); fordiv(n, d, if(!ispower(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

a(n) = A007955(n) / A183104(n).

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

cp's OEIS Frontend

A091051 Sum of divisors of n that are perfect powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A183102 a(n) = product of powerful divisors d of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A183105 a(n) = product of divisors of n that are not perfect powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula