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.

A357695 Cubefree abundant numbers.

Original entry on oeis.org

12, 18, 20, 30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 114, 126, 132, 138, 140, 150, 156, 174, 180, 186, 196, 198, 204, 210, 220, 222, 228, 234, 246, 252, 258, 260, 276, 282, 294, 300, 306, 308, 318, 330, 340, 342, 348, 350, 354, 364, 366, 372, 380, 390, 396
Offset: 1

Views

Author

Amiram Eldar, Oct 10 2022

Keywords

Comments

The least odd term is a(224) = A357697(1) = 1575.
The lower asymptotic density of this sequence is larger than 12/(91*zeta(3)) = 0.1097... which is the density of its subsequence of cubefree numbers larger than 6 and divisible by 6.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 13, 143, 1440, 14470, 144187, 1442500, 14426015, 144267400, 1442567879, 14425142573, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1442... .

Examples

			12 = 2^2 * 3 is a term since it is cubefree and sigma(12) = 28 > 2*12.

Links

Crossrefs

Intersection of A004709 and A005101.
Subsequences: A087248, A357696, A357697.
Cf. A000203 (sigma), A002117, A308618.

Programs

  • Mathematica
    f[p_, e_] := (p^(e+1)-1)/(p-1); q[1] = False; q[n_] := AllTrue[(fct = FactorInteger[n])[[;;, 2]], # < 3 &] && Times @@ f @@@ fct > 2*n; Select[Range[400], q]
  • PARI
    is(n) = {my(f = factor(n)); (n==1 || vecmax(f[,2]) < 3) && sigma(f, -1) > 2};

A357696 Cubefree primitive abundant numbers: cubefree abundant numbers having no abundant proper divisor.

Original entry on oeis.org

12, 18, 20, 30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 196, 222, 246, 258, 282, 308, 318, 354, 364, 366, 402, 426, 438, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822, 834, 836, 868, 894, 906, 942, 978, 1002
Offset: 1

Views

Author

Amiram Eldar, Oct 10 2022

Keywords

Links

Crossrefs

Intersection of A004709 and A091191.
Subsequence of A357695.
A249242 is a subsequence.
Cf. A308618.

Programs

  • Mathematica
    cubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; primQ[n_] := DivisorSigma[-1, n] > 2 && AllTrue[n/FactorInteger[n][[;; , 1]], DivisorSigma[-1, #] <= 2 &]; Select[Range[1500], cubeFreeQ[#] && primQ[#] &]
  • PARI
    is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] > 2, return(0))); if(sigma(f, -1) <= 2, return(0)); for(i = 1, #f~, if(sigma(n/f[i,1], -1) > 2, return(0))); 1};

A309875 Cubefree colossally superabundant numbers: cubefree numbers (A004709) k for which there is a positive exponent epsilon such that sigma(k)/k^{1 + epsilon} >= sigma(j)/j^{1 + epsilon} for all cubefree j > 1, so that k attains the maximum value of sigma(k)/k^{1 + epsilon} over the cubefree numbers.

Original entry on oeis.org

2, 6, 12, 60, 180, 1260, 13860, 180180, 900900, 15315300, 290990700, 6692786100, 194090796900, 6016814703900, 42117702927300, 1558355008310100, 63892555340714100, 2747379879650706300, 129126854343583196100, 6843723280209909393300, 403779673532384654204700
Offset: 1

Views

Author

Amiram Eldar, Aug 21 2019

Keywords

Comments

This sequence is formed by the largest cubefree divisors (A007948) of the colossally superabundant numbers (A004490).

Links

Crossrefs

Cf. A000203, A004490, A004709, A007948.
Subsequence of A025487, A220423 and A308618.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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