A183105 -id:A183105 - cp's OEIS frontend

A183103 a(n) = product of non-powerful divisors d of n.

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, 186624, 73

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

Sequence is not the same as A183105: a(72) = 186624, A183105(72) = 13436928.

Not multiplicative, for example 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. A001694, A007955, A183098, A183102, A183105.

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:
A183103 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if not isA001694(d) then a := a*d; end if; end do; a ; end proc:
seq(A183103(n),n=1..73) ; # 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 *)

A183103(n) = { my(m=1); fordiv(n, d, if(!ispowerful(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

a(n) = A007955(n) / A183102(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.

A183104 a(n) = product of divisors of n that are perfect powers.

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, 10368

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

Sequence is not the same as A183102: a(72) = 10368, A183102(72) = 746496.

Not multiplicative, as a(4)*a(9) <> a(36). - 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. A001597, A007955, A091051, A183102, A183105.

isA001597 := proc(n) local e ; e := seq(op(2,p),p=ifactors(n)[2]) ; return ( igcd(e) >=2 ) ; end proc:
A183104 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if isA001597(d) then a := a*d; end if; end do; a ; end proc:
seq(A183104(n),n=1..72) ; # R. J. Mathar, Jun 07 2011

perfPQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1; Table[Times@@Select[Divisors[n],perfPQ[#]&],{n,120}] (* Harvey P. Dale, Mar 07 2024 *)

A183104(n) = { my(m=1); fordiv(n, d, if(ispower(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

a(n) = A007955(n) / A183105(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.

A183101 a(n) = sum of divisors of n that are not perfect powers.

Original entry on oeis.org

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, 137, 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

Jaroslav Krizek, Dec 25 2010

nonn

Sequence is not the same as A183098(n): a(72) = 137, A183098(72) = 65.

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

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

Cf. A000203, A091051, A183098, A183105.

N:= 1000: # to get a(1) to a(N)
S:=  {seq(seq(i^k, i=1..floor(N^(1/k))), k=2..ilog2(N))}:
seq(convert(numtheory:-divisors(t) minus S,`+`), t=1..N); # Robert Israel, Oct 02 2014

Table[Total[DeleteCases[Divisors[n],?(GCD@@FactorInteger[#][[All,2]]>1&)]],{n,100}]-1 (* _Harvey P. Dale, May 30 2021 *)

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

a(n) = A000203(n) - A091051(n).

a(1) = 0, a(p) = p, a(pq) = p+q+pq, a(pq...z) = [(p+1)*(q+1)*…*(z+1)]-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

A183103 a(n) = product of non-powerful divisors d of n.

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A183104 a(n) = product of divisors of n that are perfect powers.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A183101 a(n) = sum of divisors of n that are not perfect powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula