cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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, 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

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

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

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

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

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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.
Showing 1-3 of 3 results.

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