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.

Showing 1-4 of 4 results.

A183096 a(n) = number of divisors of n that are not perfect powers.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 7, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 10, 3, 3, 3, 5, 1, 10, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

Sequence is not the same as A183093: a(72) = 7, A183093(72) = 6.

Examples

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 4.

Links

Crossrefs

Cf. A000005, A001597, A091050, A183093, A183095.
Cf. A001620, A072102.

Programs

  • Mathematica
    ppQ[n_] := GCD @@ FactorInteger[n][[;;, 2]] > 1; ppQ[1] = True; a[n_] := DivisorSum[n, 1 &, !ppQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)
  • PARI
    A091050(n) = (1+ sumdiv(n, d, ispower(d)>1)); \\ This function from Michel Marcus, Sep 21 2014
A183096(n) = (numdiv(n) - A091050(n)); \\ Antti Karttunen, Nov 23 2017

Formula

a(n) = A000005(n) - A091050(n).
a(1) = 0, a(p) = 1, a(pq) = 3, a(pq...z) = 2^k - 1, a(p^k) = 1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - A072102 - 2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 29 2025

A304817 Number of divisors of n that are either 1 or not a perfect power.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 8, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 11, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 11, 4, 4
Offset: 1

Views

Author

Gus Wiseman, May 18 2018

Keywords

Comments

First differs from A183095 at a(80) = 8, A183095(80) = 7.

Examples

			The a(72) = 8 divisors of 72 that are either 1 or not a perfect power are {1, 2, 3, 6, 12, 18, 24, 72}. Missing are {4, 8, 9, 36}.

Crossrefs

Cf. A000005, A001597, A007916, A091050, A183096, A304326, A304362, A304653, A304779, A304819, A304820.

Programs

  • Mathematica
    Table[DivisorSum[n,Boole[GCD@@FactorInteger[#][[All,2]]==1]&],{n,100}]
  • PARI
    a(n) = sumdiv(n, d, !ispower(d)); \\ Michel Marcus, May 19 2018

Formula

a(n) = A183096(n) + 1.

A183093 a(n) = number of divisors d of n such that d > 1 and if d = Product_(i) (p_i^e_i) then e_i = 1 for at least one i.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 6, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 10, 3, 3, 3, 5, 1, 10, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

a(n) = number of non-powerful divisors d of n where powerful numbers are numbers of the form A001694(m) for m >= 2.
Sequence is not the same as A183096: a(72) = 6, A183096(72) = 7.

Examples

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 4.

Links

Crossrefs

Cf. A000005, A001694, A005361, A183094, A183095, A183096.
Cf. A001620, A082695.

Programs

  • Mathematica
    nonpowQ[n_] := Min[FactorInteger[n][[;;, 2]]] == 1; a[n_] := DivisorSum[n, 1 &, # > 1 && nonpowQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)
  • PARI
    a(n) = sumdiv(n, d, d > 1 && vecmin(factor(d)[, 2]) == 1); \\ Amiram Eldar, Jan 30 2025
  • Scheme
    (define (A183093 n) (- (A000005 n) (A005361 n))) ;; Antti Karttunen, Nov 23 2017

Formula

a(n) = A000005(n) - A005361(n) = A183095(n) - 1.
a(1) = 0, a(p) = 1, a(pq) = 3, a(pq...z) = 2^k - 1, a(p^k) = 1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - zeta(2)*zeta(3)/zeta(6) - 1), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 30 2025

Extensions

Name corrected by Amiram Eldar, Jan 30 2025

A183094 a(n) = number of powerful divisors d (excluding 1) of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 1, 3, 0, 0, 0, 2, 0
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

a(n) = number of divisors d of n from set A001694(m) - powerful numbers for m >=2.

Examples

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

Links

Crossrefs

Cf. A000005, A001694, A005361, A082695, A183095.

Programs

  • Maple
    f:=  n -> convert(map(t->t[2], ifactors(n)[2]),`*`) - 1; # Robert Israel, Jul 14 2014
  • Mathematica
    powerfulQ[n_] := Min[ Last@# & /@ FactorInteger[n]] > 1; f[n_] := Length@ Select[ Divisors@ n, powerfulQ]; Array[f, 105] (* Robert G. Wilson v, Jul 14 2014 *)

Formula

a(n) = A000005(n) - A183095(n) = A005361(n) - 1.
a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = k-1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)*zeta(3)/zeta(6) - 1 = A082695 - 1 = 0.9435964368... . - Amiram Eldar, Jul 30 2022
Showing 1-4 of 4 results.

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