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.

A357686 Nonsquarefree numbers k such that A293228(k) > k.

Original entry on oeis.org

60, 84, 132, 140, 156, 204, 228, 276, 348, 372, 420, 444, 492, 516, 564, 636, 660, 708, 732, 780, 804, 852, 876, 924, 948, 996, 1020, 1068, 1092, 1140, 1164, 1212, 1236, 1284, 1308, 1356, 1380, 1428, 1524, 1540, 1572, 1596, 1644, 1668, 1716, 1740, 1788, 1812, 1820
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2022

Keywords

Comments

The squarefree numbers k such that A293228(k) > k are the squarefree abundant numbers (A087248).
If k > 3 is a term of A243128 then 4*k is a term.
The least odd term is (3/2)*prime(17)# = 2884140525231318958605.
The least term that is coprime to 6 is (5/6)*prime(1245)# = 5.629...*10^4361.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 2, 26, 287, 2725, 27660, 275298, 2754638, 27556849, 275538900, 2755151247, ... . Apparently, the asymptotic density of this sequence exists and equals 0.02755... .

Examples

			60 = 2^2 * 15 is a term since it is nonsquarefree, its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15, 30} and their sum is 72 > 60.

Links

Crossrefs

Intersection of A013929 and A357685.
Subsequence of A005101.
Cf. A087248, A243128.

Programs

  • Mathematica
    q[n_] := AnyTrue[(f = FactorInteger[n])[[;;, 2]], # > 1 &] && Times @@ (1 + f[[;; , 1]]) > n; Select[Range[2, 2000], q]
  • PARI
    is(n) = {my(f = factor(n)); if(n == 1 || vecmax(f[,2]) == 1, return(0)); prod(i=1, #f~, f[i,1]+1) > n};

A357685 Numbers k such that A293228(k) > k.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 102, 114, 132, 138, 140, 156, 174, 186, 204, 210, 222, 228, 246, 258, 276, 282, 318, 330, 348, 354, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564, 570, 582, 606, 618, 636, 642, 654, 660, 678, 690
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2022

Keywords

Comments

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 7, 79, 843, 8230, 83005, 826875, 8275895, 82790525, 827718858, 8276571394, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0827... .

Examples

			30 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15} and their sum is 42 > 30.
60 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15, 30} and their sum is 72 > 60.

Links

Crossrefs

Cf. A005117, A293228.
Disjoint union of A087248 and A357686.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

Programs

  • Mathematica
    s[n_] := Times @@ (1 + (f = FactorInteger[n])[[;; , 1]]) - If[AllTrue[f[[;;, 2]], # == 1 &], n, 0]; Select[Range[2, 1000], s[#] > # &]
  • PARI
    is(n) = {my(f = factor(n), s); s = prod(i=1, #f~, f[i,1]+1); if(n==1 || vecmax(f[,2]) == 1, s -= n); s > n};

A293227 a(n) is the number of proper divisors of n that are squarefree.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 7, 1, 2, 3, 3, 3, 4, 1, 3, 3, 4, 1, 7, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 8, 1, 3, 4, 2, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 4, 4, 3, 7, 1, 4, 2, 3, 1, 8, 3, 3, 3, 4, 1, 8, 3, 4, 3, 3, 3, 4, 1, 4, 4, 4, 1, 7, 1, 4, 7
Offset: 1

Views

Author

Antti Karttunen, Oct 08 2017

Keywords

Links

Crossrefs

Cf. A001221, A005117, A008683, A008966, A034444, A293228, A293234.
Cf. A001620, A306016.

Programs

Formula

a(n) = Sum_{d|n, dA008966(d).
a(n) = A034444(n) - A008966(n).
a(n) = 2^A001221(n) - A008683(n)^2 = 2^omega(n) - mu(n)^2.
G.f.: Sum_{k>=1} mu(k)^2*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Oct 28 2018
Sum_{k=1..n} a(k) ~ (6/Pi^2)*n*(log(n) + 2*(gamma - 1 - zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A293235 a(n) is the sum of proper divisors of n that are square.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 14, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 1, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 21, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 21, 1, 50, 10, 30, 1, 1, 1, 5, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 08 2017

Keywords

Comments

a(n) = 1 if and only if n > 1 is squarefree or the square of a prime. - Robert Israel, Oct 08 2017

Links

Crossrefs

Cf. A010052, A035316, A078434, A293228, A293234.

Programs

  • Maple
    A035316:= n -> mul((p[1]^(p[2]+2-(p[2] mod 2))-1)/(p[1]^2-1), p = ifactors(n)[2]):
f:= n -> A035316(n) - `if`(issqr(n),n,0):
map(f, [$1..100]); # Robert Israel, Oct 08 2017
  • Mathematica
    Table[Total[Select[Most[Divisors[n]],IntegerQ[Sqrt[#]]&]],{n,120}] (* Harvey P. Dale, Dec 29 2023 *)
  • PARI
    A293235(n) = sumdiv(n,d,(d

Formula

a(n) = Sum_{d|n, dA010052(d)*d.
a(n) = A035316(n) - (A010052(n)*n).
G.f.: Sum_{k>=1} k^2 * x^(2*k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 13 2021
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3/2)-1)/3 = 0.537458449561... . - Amiram Eldar, Dec 01 2023

A357698 a(n) is the sum of the aliquot divisors of n that are cubefree.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 7, 1, 21, 1, 22, 11, 14, 1, 28, 6, 16, 13, 28, 1, 42, 1, 7, 15, 20, 13, 55, 1, 22, 17, 42, 1, 54, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 39, 17, 56, 23, 32, 1, 108, 1, 34, 41, 7, 19, 78, 1, 58, 27, 74, 1, 91, 1, 40
Offset: 1

Views

Author

Amiram Eldar, Oct 10 2022

Keywords

Examples

			The divisors of 16 that are cubefree are {1, 2, 4}, and their sum is a(16) = 1 + 2 + 4 = 7.

Links

Crossrefs

Cf. A004709, A073185, A212793, A293228.
Cf. A013661, A002117.

Programs

  • Mathematica
    f[p_, e_] := 1 + p + If[e == 1, 0, p^2]; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;;, 2]], # < 3 &], n, 0]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, 1 + f[i,1] + if(f[i,2] == 1, 0, f[i,1]^2)); if(n==1 || vecmax(f[,2]) < 3, s -= n); s};

Formula

a(n) = Sum_{d|n, dA212793(d)*d.
a(n) = A073185(n) - (A212793(n)*n).
a(n) = 1 iff n is a prime.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2) - 1)/(2*zeta(3)) = 0.268262... .
Showing 1-5 of 5 results.

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