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.

A308053 Coreful abundant numbers: numbers k such that csigma(k) > 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

72, 108, 144, 200, 216, 288, 324, 360, 400, 432, 504, 540, 576, 600, 648, 720, 756, 784, 792, 800, 864, 900, 936, 972, 1000, 1008, 1080, 1152, 1188, 1200, 1224, 1296, 1368, 1400, 1404, 1440, 1512, 1568, 1584, 1600, 1620, 1656, 1728, 1764, 1800, 1836, 1872, 1936
Offset: 1

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Author

Amiram Eldar, May 10 2019

Keywords

Comments

Analogous to A005101 as A307958 is analogous to A000396.
The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 0.0262215..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, Sep 02 2022

Examples

			72 is in the sequence since its coreful divisors are 6, 12, 18, 24, 36, 72, whose sum is 168 > 2 * 72.

Links

Crossrefs

Cf. A005101, A057723, A307958.
A339940 and A356871 are subsequences.
Subsequence of A129575.

Programs

  • Mathematica
    f[p_, e_] := (p^(e+1)-1)/(p-1)-1; a[1]=1; a[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[If[a[n] > 2n, AppendTo[s, n]], {n, 1, 2000}]; s
  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
isok(k) = s(k) > 2*k; \\ Michel Marcus, May 11 2019
  • PARI
    isok(k) = {my(f=factor(k)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1)-1) > 2*k}; \\ Amiram Eldar, Sep 02 2022

A363169 Powerful abundant numbers: numbers that are both powerful (A001694) and abundant (A005101).

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 784, 800, 864, 900, 968, 972, 1000, 1152, 1296, 1352, 1372, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2500, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000
Offset: 1

Views

Author

Amiram Eldar, May 19 2023

Keywords

Comments

The least odd term is a(90) = 11025, and the least term that is coprime to 6 is 1382511906801025.
Are there two consecutive integers in this sequence? There are none below 10^22.

Examples

			36 = 2^2 * 3^2 is a term since it is powerful, and sigma(36) = 91 > 2*36 = 72.

Links

Crossrefs

Intersection of A001694 and A005101.
Cf. A180114, A363170, A363171.
Subsequences: A307959, A328136, A356871.

Programs

  • Mathematica
    Select[Range[4000], DivisorSigma[-1, #] > 2 && Min[FactorInteger[#][[;;, 2]]] > 1 &]
  • PARI
    is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; }

A339936 Odd coreful abundant numbers: the odd terms of A308053.

Original entry on oeis.org

99225, 165375, 231525, 297675, 496125, 694575, 826875, 893025, 1091475, 1157625, 1225125, 1289925, 1488375, 1620675, 1686825, 1819125, 1885275, 2083725, 2149875, 2282175, 2480625, 2546775, 2679075, 2811375, 2877525, 3009825, 3075975, 3142125, 3274425, 3472875
Offset: 1

Views

Author

Amiram Eldar, Dec 23 2020

Keywords

Comments

The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 9.1348...*10^(-6), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022

Examples

			99225 is a term since it is odd and the sum of its coreful divisors is A057723(99225) = 201600 > 2 * 99225.

Links

Crossrefs

Intersection of A005408 and A308053.
Subsequence of A321147.
Cf. A057723, A356871.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A293186, A321147, A333950.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[1, 10^6, 2], s[#] > 2*# &]

A364991 Primitive coreful 3-abundant numbers: coreful 3-abundant numbers (A340109) that are powerful numbers (A001694).

Original entry on oeis.org

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 42336, 43200, 48600, 54000, 56448, 57600, 63504, 64800, 72000, 81000, 84672, 86400, 88200, 90000, 97200, 98784, 104544, 108000, 112896, 115200, 127008, 129600, 135000, 144000, 145800
Offset: 1

Views

Author

Amiram Eldar, Aug 15 2023

Keywords

Comments

Powerful numbers k such that csigma(k) > 3*k, where csigma(k) = A057723(k) is the sum of the coreful divisors of k.
If m is a term and k is a squarefree number coprime to m, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 3*k*m, so k*m is a coreful 3-abundant number. Therefore, the sequence of coreful 3-abundant numbers (A340109) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful 3-abundant numbers can be calculated from this sequence (see comment in A340109).

Links

Crossrefs

Intersection of A001694 and A340109.
Subsequence of A356871.

Programs

  • Mathematica
    f[p_, e_] := (p^(e+1)-1)/(p-1)-1; g[1] = 1; g[n_] := If[AllTrue[(fct = FactorInteger[n])[[;; , 2]], #>1 &], Times @@ f @@@ fct, 0]; seq[kmax_] := Module[{s = {}}, Do[If[g[k] > 3*k, AppendTo[s, k]], {k, 1, kmax}]; s]; seq[500000]
  • PARI
    s(f) = prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);
lista(kmax) = {my(f); for(k=2, kmax, f=factor(k); if(vecmin(f[,2]) > 1 && s(f) > 3*k, print1(k, ", ")));}
Showing 1-4 of 4 results.

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