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 1-3 of 3 results.

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

A183095 a(n) = number of divisors d of n that are either 1 or of the form Product_(i) (p_i^e_i) where e_i = 1 for at least one i.

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, 7, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 11, 4, 4, 4, 6, 2, 11, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6
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 >= 1.

Examples

			For n = 12, set of such divisors is {1, 2, 3, 6, 12}; a(12) = 5.

Links

Crossrefs

Cf. A000005, A001694, A183093, A183094.
Cf. A001620, A082695.

Programs

  • Mathematica
    nonpowQ[n_] := Min[FactorInteger[n][[;;, 2]]] == 1;a[n_] := DivisorSum[n, 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

Formula

a(n) = A000005(n) - A183094(n) = A183093(n) + 1.
a(1) = 1, a(p) = 2, a(pq) = 4, a(pq...z) = 2^k, a(p^k) = 2, 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)), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 30 2025

Extensions

Name corrected by Amiram Eldar, Jan 30 2025

A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).

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, 4, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0
Offset: 1

Views

Author

Amiram Eldar, Dec 19 2023

Keywords

Comments

First differs from A046660 and A066301 at n = 36, and from A183094 at n = 72.
Let b(n, k) be the sequence that counts the divisors of n that are k-th powers of squarefree numbers. Then, b(n, 1) = A034444(n), b(n, 2) = A323308(n), b(n, 3) = A368248(n). b(n, k) is multiplicative with b(p^e, k) = 2 if e >= k, and 1 otherwise. The asymptotic mean of b(n, k) for k >= 2 is lim_{m->oo} (1/m) * Sum_{n=1..m} b(n, k) = zeta(k)/zeta(2*k). Since a(n) = Sum_{k>=2} (b(n, k) - 1), the formula for the asymptotic mean of this sequence follows (see the Formula section).

Links

Crossrefs

Cf. A005117, A048105, A072777, A327527, A368250.
Cf. A034444, A323308, A368248.
Cf. A046660, A066301, A183094.

Programs

  • Mathematica
    a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1] - 2^Length[e]]; a[1] = 0; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), e, m, h, c); if(n == 1, 0, e = f[,2]; m = vecmax(e); h = vector(m); for(i = 1,m, c = 0; for(j = 1, #e, if(e[j] == (m+1-i), c++)); h[i] = c); for(i = 2, m, h[i] += h[i-1]); for(i = 1, m, h[i] = 2^h[i]-1); 1 + vecsum(h) - 1<<#e);}

Formula

a(n) = A327527(n) - A034444(n).
a(n) = 0 if and only if n is squarefree (A005117).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) = 0.848633... (A368250).
Showing 1-3 of 3 results.

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