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 1-8 of 8 results.

A231086 Initial members of abundant twins, i.e., values of k such that (k, k+2) are both abundant numbers.

Original entry on oeis.org

18, 40, 54, 70, 78, 88, 100, 102, 112, 138, 160, 174, 196, 198, 208, 220, 222, 258, 270, 280, 304, 306, 318, 340, 348, 350, 352, 364, 366, 378, 390, 400, 414, 438, 448, 460, 462, 474, 490, 498, 520, 532, 544, 550, 558, 570, 580, 606, 616, 618, 640, 642, 648
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 03 2013

Keywords

Comments

The first odd term is <= 76728582876430878992529528245373 (see A294025). Note that there are infinitely many odd terms, since if k is an odd term then 2*t*k*(k+2) + k are odd terms for all t >= 0. - Jianing Song, Nov 13 2022
From Amiram Eldar, May 30 2023: (Start)
The least odd term is larger than 10^11.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 7, 81, 820, 8074, 80410, 804623, 8040362, 80414534, 804257458, 8042148484, ... . Apparently, the asymptotic density of this sequence exists and equals 0.08042... . (End)

Examples

			18, 20 are abundant, thus the smaller number is listed.

Links

Crossrefs

Cf. A005101, A096399, A108926, A231088, A231089, A231090, A231092, A231093, A303741.

Programs

  • GAP
    A:=Filtered([1..700],n->Sigma(n)>2*n);;  a:=List(Filtered([1..Length(A)-1],i->A[i+1]=A[i]+2),j->A[j]); # Muniru A Asiru, Jun 24 2018
  • Maple
    withnumtheory: select(n->sigma(n)>2*n and sigma(n+1)<2*(n+1) and sigma(n+2)>2*(n+2),[$1..700]); # Muniru A Asiru, Jun 24 2018
  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2n; m = 0; a2 = {}; Do[If[AbundantQ[n], m = m + 1; If[m > 1, AppendTo[a2, n - 2]], m = 0], {n, 2, 100000, 2}];a2
Module[{nn=650,sa},sa=Table[If[DivisorSigma[1,n]>2n,1,0],{n,nn}];Transpose[ SequencePosition[sa,{1,0,1}]]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, May 20 2016 *)
  • PARI
    is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

Formula

a(n) = A005101(A303741(n)). - Amiram Eldar, Mar 01 2025

A108926 Initial members of abundant quintuplets, i.e., values of k such that (k, k+2, k+4, k+6, k+8) are all abundant numbers.

Original entry on oeis.org

2988, 4728, 9724, 18844, 22984, 30544, 35148, 39948, 45048, 50464, 55788, 56808, 58056, 58780, 69184, 78048, 81948, 85744, 101148, 106144, 108256, 109248, 117124, 134088, 139744, 139804, 152568, 171288, 174348, 175908, 182644, 189768, 197028
Offset: 1

Views

Author

Jason Earls, Jul 19 2005

Keywords

Links

Crossrefs

Cf. A005101, A096399, A231086, A231089, A231090, A231092, A231093.

Programs

  • Mathematica
    SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,200000}],{1,,1,,1,,1,,1}][[All,1]] (* Harvey P. Dale, Mar 06 2022 *)
  • PARI
    is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 && sigma(n+4,-1)>2 && sigma(n+6,-1)>2 && sigma(n+8,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

A231088 Initial members of abundant triples, i.e., values of k such that (k, k+2, k+4) are all abundant numbers.

Original entry on oeis.org

100, 196, 220, 304, 348, 350, 364, 460, 616, 640, 700, 736, 832, 1036, 1060, 1144, 1180, 1216, 1312, 1372, 1456, 1480, 1660, 1696, 1876, 1900, 1936, 1984, 1998, 2000, 2020, 2176, 2208, 2210, 2296, 2320, 2548, 2620, 2716, 2740, 2748, 2750, 2988, 2990, 2992
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 03 2013

Keywords

Examples

			100, 102, 104 are abundant, thus the smallest number is listed.

Links

Crossrefs

Cf. A005101, A096399, A108926, A231086, A231089, A231090, A231092, A231093.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2n; m = 0; a = {}; Do[If[AbundantQ[n], m = m + 1; If[m > 2, AppendTo[a, n - 4]], m = 0], {n, 2, 1000000, 2}];a
2*Flatten[Position[Partition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,2,3000,2}],3,1], {1,1,1}]] (* Harvey P. Dale, Aug 19 2014 *)
2*SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,2,3000,2}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Feb 27 2023 *)
  • PARI
    is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 && sigma(n+4,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

A231089 Initial members of abundant quadruplets, i.e., values of k such that (k, k+2, k+4, k+6) are all abundant numbers.

Original entry on oeis.org

348, 1998, 2208, 2748, 2988, 2990, 3006, 3246, 3708, 3846, 4506, 4728, 4730, 5166, 6228, 7068, 7206, 7908, 8886, 9348, 9588, 9724, 9726, 11406, 13746, 14208, 14766, 17148, 17988, 18126, 18588, 18828, 18844, 18846, 19548, 20148, 20478, 21486, 22188, 22984
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 03 2013

Keywords

Examples

			348, 350, 352, 354 are abundant, thus the smallest number is listed.

Links

Crossrefs

Cf. A005101, A096399, A108926, A231086, A231088, A231090, A231092, A231093.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2n; m = 0; a = {}; Do[If[AbundantQ[n], m = m + 1; If[m > 3, AppendTo[a, n - 6]], m = 0], {n, 2, 1000000, 2}];a
SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,23000}],{1,,1,,1,,1}][[All,1]] (* _Harvey P. Dale, Apr 02 2018 *)
  • PARI
    is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 && sigma(n+4,-1)>2 && sigma(n+6,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

A231090 Initial members of abundant sextuplets, i.e., values of n such that (n, n+2, n+4, n+6, n+8, n+10) are all abundant numbers.

Original entry on oeis.org

801340, 962650, 7353340, 21964300, 41642140, 48049690, 55455940, 89034940, 89851450, 92253850, 105259540, 107948380, 109455340, 114295450, 116754940, 122349370, 135575980, 156009850, 159521050, 173645440, 188586700, 192674170, 193137850, 220301770, 221355126
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 03 2013

Keywords

Examples

			801340, 801342, 801344, 801346, 801348, 801350 are abundant, thus the smallest number is listed.

Links

Crossrefs

Cf. A005101, A108926, A231086, A231088, A231089, A231092, A231093.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2n; m = 0; a = {}; Do[If[AbundantQ[n], m = m + 1; If[m > 5, AppendTo[a, n - 10]], m = 0], {n, 2, 1000000000, 2}];a
2*SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,2,2214*10^5,2}],{1,1,1,1,1,1}][[All,1]] (* Harvey P. Dale, May 12 2022 *)
  • PARI
    is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 && sigma(n+4,-1)>2 && sigma(n+6,-1)>2 && sigma(n+8,-1)>2 && sigma(n+10,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

A231092 Initial members of abundant septuplets, i.e., values of n such that (n, n+2, n+4, n+6, n+8, n+10, n+12) are all abundant numbers.

Original entry on oeis.org

221355126, 221355128, 402640540, 402640542, 668862580, 668862582, 739577140, 739577142, 1415514246, 1415514248, 1598558646, 1598558648, 1678915540, 1678915542, 1714512246, 1714512248, 1812156340, 1812156342, 1829740086, 1829740088, 1892686326, 1892686328
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 03 2013

Keywords

Comments

If the terms are divided into groups of two, are the differences between the two grouped terms always 2? - Harvey P. Dale, Mar 24 2025

Examples

			221355126, 221355128, 221355130, 221355132, 221355134, 221355136, 221355138 are abundant, thus the smallest number is listed.

Links

Crossrefs

Cf. A005101, A108926, A231086, A231088, A231089, A231090, A231093.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2n; m = 0; a = {}; Do[If[AbundantQ[n], m = m + 1; If[m > 6, AppendTo[a, n - 12]], m = 0], {n, 2, 2000000000, 2}];a
SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,18927*10^5}],{1,,1,,1,,1,,1,,1,,1}][[;;,1]] (* Harvey P. Dale, Mar 24 2025 *)
  • PARI
    is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 && sigma(n+4,-1)>2 && sigma(n+6,-1)>2 && sigma(n+8,-1)>2 && sigma(n+10,-1)>2 && sigma(n+12,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

A228965 Smallest sets of 8 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed.

Original entry on oeis.org

221355126, 402640540, 668862580, 739577140
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 10 2013

Keywords

Comments

Is this a duplicate of A231093? - R. J. Mathar, Nov 15 2013
No; there is some element of this sequence not in A231093 below approximately 10^10^72. In fact A228965 \ A231093 has positive lower density (though presumably quite small). Capsule proof: choose n such that an appropriately large number of primes divide n, n+1, ..., n+7. Since the reciprocal of the primes diverges, you can get sigma(n+i)/(n+i) arbitrarily large. - Charles R Greathouse IV, Nov 15 2013

Examples

			221355126, 221355128, 221355130, 221355132, 221355134, 221355136, 221355138, 221355140  is the smallest set of 8 consecutive abundant numbers in arithmetic progression so 221355126 is in the list.

Crossrefs

Cf. A005101, A228433, A228844, A228961, A228962, A228963, A228964.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2 n; m = 2; z1 = 18; cd = 6; a = {}; Do[If[AbundantQ[n], If[n - z1 == cd, m = m + 1; If[m > 7, AppendTo[a, n - 7*cd]], m = 2; cd = n - z1]; z1 = n], {n, 19, 1000000000}]; a

A231094 Smallest starting abundant number for n consecutive even abundant numbers.

Original entry on oeis.org

12, 18, 100, 348, 2988, 801340, 221355126, 221355126, 895257140404
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 03 2013

Keywords

Examples

			a(6) = 801340 because 801340 is the starting abundant number for 6 consecutive even abundant numbers: 801340, 801342, 801344, 801346, 801348, 801350.

Crossrefs

Cf. A005101, A108926, A231086, A231088, A231089, A231090, A231092, A231093.

Programs

  • Mathematica
    k = 2; cnt = 0; Table[While[If[abunQ[k], cnt++, cnt = 0]; cnt < n, k = k + 2]; k = k + 2; k - 2*n, {n, 6}] (* T. D. Noe, Nov 04 2013 *)

Extensions

a(9) from Donovan Johnson, Nov 07 2013
Showing 1-8 of 8 results.

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