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 1-5 of 5 results.

A368038 The sum of non-unitary divisors of the nonsquarefree numbers.

Original entry on oeis.org

2, 6, 3, 8, 14, 9, 12, 24, 5, 12, 16, 30, 41, 36, 24, 18, 56, 7, 15, 28, 36, 48, 48, 24, 62, 36, 105, 20, 40, 84, 39, 64, 72, 54, 48, 120, 21, 36, 87, 84, 140, 112, 60, 42, 144, 11, 64, 30, 72, 126, 96, 72, 108, 96, 233, 28, 76, 60, 120, 54, 112, 180, 117, 84
Offset: 1

Views

Author

Amiram Eldar, Dec 09 2023

Keywords

Comments

The positive terms of A048146, since A048146(k) = 0 if and only if k is squarefree (A005117).

Links

Crossrefs

Cf. A005117, A048146, A013929.
Cf. A084936, A174961, A275699, A368039, A368040.
Cf. A002117, A013661, A072691, A229099.

Programs

  • Mathematica
    f[p_, e_] := (p^(e+1)-1)/(p-1); nusigma[n_] := Module[{fct = FactorInteger[n]}, If[n == 1, 0, Times @@ f @@@ fct - Times @@ (1 + Power @@@ fct)]]; Select[Array[nusigma, 200], # > 0 &]
  • PARI
    lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(sigma(f) - prod(i=1, #f~, 1+f[i,1]^f[i,2]), ", ")));}

Formula

a(n) = A048146(A013929(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/2)*(1-1/zeta(3))/(1-1/zeta(2))^2 = 0.899359898779... .

A368040 The powerful part of the nonsquarefree numbers.

Original entry on oeis.org

4, 8, 9, 4, 16, 9, 4, 8, 25, 27, 4, 32, 36, 8, 4, 9, 16, 49, 25, 4, 27, 8, 4, 9, 64, 4, 72, 25, 4, 16, 81, 4, 8, 9, 4, 32, 49, 9, 100, 8, 108, 16, 4, 9, 8, 121, 4, 125, 9, 128, 4, 27, 8, 4, 144, 49, 4, 25, 8, 9, 4, 32, 81, 4, 8, 169, 9, 4, 25, 16, 36, 8, 4, 27
Offset: 1

Views

Author

Amiram Eldar, Dec 09 2023

Keywords

Comments

The terms of A057521 that are larger than 1, since A057521(k) = 1 if and only if k is squarefree (A005117).

Links

Crossrefs

Cf. A005117, A013929, A057521.
Cf. A084936, A174961, A275699, A368038, A368039.
Cf. A013661, A229099, A328013.

Programs

  • Mathematica
    f[p_, e_] := If[e > 1, p^e, 1]; powPart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[powPart, 200], # > 1 &]
  • PARI
    lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); if(p > 1, print1(p, ", ")));}

Formula

a(n) = A057521(A013929(n)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = d/(3*(1-1/zeta(2))^(3/2)) = 4.778771..., and d = A328013.

A368541 The number of exponential divisors of the nonsquarefree numbers.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 4, 2, 2, 2, 4, 4, 2, 4, 2, 2, 3, 2, 4, 2, 2, 4, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 3
Offset: 1

Views

Author

Amiram Eldar, Dec 29 2023

Keywords

Comments

The terms of A049419 that are larger than 1, since A049419(k) = 1 if and only if k is squarefree (A005117).

Links

Crossrefs

Cf. A005117, A013929, A049419.
Cf. A084936, A174961, A275699, A368038, A368039, A368040.
Cf. A013661, A059956, A229099, A327837.

Programs

  • Mathematica
    f[p_, e_] := DivisorSigma[0, e]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 1 &]
  • PARI
    lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, numdiv(f[i, 2])); if(p > 1, print1(p, ", ")));}

Formula

a(n) = A049419(A013929(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (A327837 - A059956)/A229099 = 2.53623753427906735929... .

A382421 The product of exponents in the prime factorization of the noncubefree numbers.

Original entry on oeis.org

3, 4, 3, 3, 5, 3, 4, 3, 3, 6, 6, 4, 4, 3, 5, 3, 6, 4, 3, 3, 7, 3, 3, 8, 3, 5, 4, 3, 4, 3, 3, 6, 6, 4, 9, 5, 3, 4, 5, 3, 3, 8, 3, 3, 4, 3, 10, 3, 3, 4, 3, 6, 8, 3, 4, 3, 3, 3, 5, 6, 4, 3, 3, 3, 7, 6, 8, 4, 3, 5, 3, 12, 3, 6, 3, 3, 4, 3, 5, 5, 3, 4, 6, 6, 9, 3, 3
Offset: 1

Views

Author

Amiram Eldar, Mar 25 2025

Keywords

Links

Crossrefs

Cf. A002117, A013661, A013664, A005361, A046099, A082695, A330594, A368039, A382419.

Programs

  • Mathematica
    s[n_] := Times @@ FactorInteger[n][[;; , 2]]; noncubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] > 2; s /@ Select[Range[600], noncubeFreeQ]
  • PARI
    list(kmax) = {my(e); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) > 2, print1(vecprod(e), ", "))); }

Formula

a(n) = A005361(A046099(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)*zeta(3)^2/zeta(6) - zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3))/(zeta(3) - 1) = (A082695 - A330594) * A002117 / (A002117 - 1) = 4.97723390794900554553... .

A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

Original entry on oeis.org

2, 6, 3, 6, 14, 6, 10, 18, 5, 12, 14, 30, 36, 30, 22, 15, 42, 7, 10, 26, 24, 42, 30, 21, 62, 34, 96, 15, 38, 70, 39, 42, 66, 30, 46, 90, 14, 33, 80, 78, 126, 98, 58, 39, 90, 11, 62, 30, 42, 126, 66, 60, 102, 70, 216, 21, 74, 30, 114, 51, 78, 150, 78, 82, 126, 13
Offset: 1

Views

Author

Amiram Eldar, May 21 2024

Keywords

Comments

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
The positive terms of A336563: if k is a squarefree number (A005117) then the only coreful divisor of k is k itself, so k has no aliquot coreful divisors.
The number of the aliquot coreful divisors of the n-th nonsquarefree number is A368039(n).

Links

Crossrefs

Cf. A005117, A013661, A013929, A065487, A229099, A307958, A336563.
Cf. A084936, A174961, A275699, A368038, A368039, A368040, A368541, A368713.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; Select[Array[s, 300], # > 0 &]
  • PARI
    lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - 1) - k, ", "))); }
  • Python
    from math import prod, isqrt
from sympy import mobius, factorint
def A373058(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod((p**(e+1)-1)//(p-1)-1 for p, e in factorint(m).items())-m # Chai Wah Wu, Jul 22 2024

Formula

a(n) = A336563(A013929(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (A065487 - 1)/(1-1/zeta(2))^2 = 1.50461493205911656114... .
Showing 1-5 of 5 results.

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