A231086 -id:A231086 - cp's OEIS frontend

A231093 Initial members of abundant octuplets, i.e., values of n such that (n, n+2, n+4, n+6, n+8, n+10, n+12, n+14) are all abundant numbers.

221355126, 402640540, 668862580, 739577140, 1415514246, 1598558646, 1678915540, 1714512246, 1812156340, 1829740086, 1892686326, 2097915966, 2259080046, 2452774780, 2453605540, 2521418740, 2726361940, 3118553740, 3252749646, 3318076446, 4119153340, 4748101660

Offset: 1

Shyam Sunder Gupta, Nov 03 2013

nonn

			221355126, 221355128, 221355130, 221355132, 221355134, 221355136, 221355138, 221355140 are abundant, thus the smallest number is listed.

Donovan Johnson, Table of n, a(n) for n = 1..500

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

 AbundantQ[n_] := DivisorSigma[1, n] > 2n; m = 0; a = {}; Do[If[AbundantQ[n], m = m + 1; If[m > 7, AppendTo[a, n - 14]], m = 0], {n, 2, 2000000000, 2}];a

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

Jason Earls, Jul 19 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Shyam Sunder Gupta)

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

SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,200000}],{1,,1,,1,,1,,1}][[All,1]] (* Harvey P. Dale, Mar 06 2022 *)

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

Shyam Sunder Gupta, Nov 03 2013

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Shyam Sunder Gupta)

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

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 *)

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

Shyam Sunder Gupta, Nov 03 2013

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Shyam Sunder Gupta)

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

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 *)

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

Shyam Sunder Gupta, Nov 03 2013

nonn

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

Shyam Sunder Gupta and Donovan Johnson, Table of n, a(n) for n = 1..5000 (first 126 terms from Shyam Sunder Gupta)

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

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 *)

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

Shyam Sunder Gupta, Nov 03 2013

nonn

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

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

Donovan Johnson, Table of n, a(n) for n = 1..1000

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

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 *)

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

A303741 Numbers k such that A(k+1) = A(k) + 2, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

2, 7, 10, 14, 16, 19, 22, 23, 26, 31, 36, 39, 44, 45, 48, 51, 52, 59, 62, 65, 70, 71, 74, 79, 81, 82, 83, 86, 87, 90, 93, 96, 99, 104, 107, 110, 111, 114, 118, 120, 125, 128, 131, 133, 135, 138, 141, 146, 149, 150, 155, 156, 158, 164, 169, 170, 175, 178, 179

Offset: 1

Muniru A Asiru, Jun 22 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Muniru A Asiru)

A231086 is the main entry for this sequence.

Cf. A005101, A091194, A096399, A169822.

A:=Filtered([1..1000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+2);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..1000]):  a:=select(j->A[j+1]=A[j]+2,[$1..nops(A)-1]);

Position[Differences[Select[Range[750], DivisorSigma[1, #] > 2*# &]], 2] // Flatten (* Amiram Eldar, Mar 15 2024 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 2, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Sequence is { k | A005101(k+1) = A005101(k) + 2 }.

a(n) = A091194(A231086(n)). - Amiram Eldar, Mar 01 2025

A306952 Lesser member of twin weird numbers: weird numbers n (A006037) such that n+2 is also weird.

Original entry on oeis.org

512468, 540890, 688028, 1390268, 1565828, 1741388, 2268068, 3525410, 3848108, 4374788, 6481508, 6657068, 7534868, 7885988, 7914410, 8089970, 8838968, 9143330, 9290468, 10021130, 10343828, 10898930, 12654530, 12801668, 12872510, 13152788, 13181210, 14234570

Offset: 1

Amiram Eldar, Mar 17 2019

nonn

Number of terms below 10^k for k = 6, 7, ... 10: 19, 231, 2111, 22426.

The first occurrences of 2 consecutive pairs of twin weirds are (21607670, 21607672, 21608090, 21608092), (873951608, 873951610, 873951890, 873951892), ...

			512468 is in the sequence since both 512468 and 512470 are weird numbers.

Amiram Eldar, Table of n, a(n) for n = 1..22426 (terms below 10^10).

Cf. A006037, A125109, A231086 (supersequence), A231964.

A306497 Abundant numbers that differ from the next abundant number by 5.

Original entry on oeis.org

5391411025, 26957055120, 28816162375, 33426748350, 34393484125, 37739877175, 40342627320, 48150877770, 50866790970, 53356378075, 59305521270, 60711976320, 61164628525, 63395557225, 64899009175, 67275433225, 70088343325, 74922022170, 75665665075, 76781129425

Offset: 1

Sergio Pimentel, Feb 19 2019

nonn

Since all multiples of 6 are abundant, numbers in this sequence have to be abundant numbers of the form 6n or 6n + 1. Most common difference between abundant numbers is 6, followed by 2, 4, 3, 1. 5 is the least common.

			a(1) = 5391411025 is in the sequence since it is abundant and the next abundant number is 5391411030 which is a(1)+5 and all the numbers in between are deficient.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005101, A096399, A231086, A228382, A316098, A316099, A115414, A125115.

isok(n) = for(k=1, 4, if(sigma(n+k) > 2*(n+k), return(0))); (sigma(n) > 2*n) && (sigma(n+5) > 2*(n+5)); \\ Daniel Suteu, Jul 24 2019

Either a(n) or a(n)+5 are in A115414. - Amiram Eldar, Jul 16 2019

More terms from Amiram Eldar, Jul 16 2019

A329525 a(n) is the smallest positive number k such that k and k+n are both abundant.

Original entry on oeis.org

5775, 18, 942, 20, 940, 12, 945, 12, 936, 20, 4725, 12, 4712, 40, 930, 20, 928, 12, 2816, 20, 924, 18, 945, 12, 920, 30, 918, 12, 2176, 12, 3465, 24, 912, 20, 910, 12, 7208, 18, 906, 20, 4095, 12, 5312, 12, 900, 20, 945, 12, 896, 20, 894, 18, 4672, 12, 945, 24

Offset: 1

Jaroslav Krizek, Nov 15 2019

nonn

Sequences of numbers k such that k and k+n are both abundant for any n: A096399 (n = 1), A231086 (n = 2).

			Number 5775 is the smallest abundant number k such that k+1 = 5576 is also abundant.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005101, A096399, A231086, A294937.

[Min([m: m in[1..10^4] | SumOfDivisors(m) gt 2*m and SumOfDivisors(m+n) gt 2*(m+n)]): n in [1..60]];

A329525(n) = for(k=1, oo, if((sigma(k) > (k+k)) && (sigma(n+k) > 2*(n+k)), return(k))); \\ Antti Karttunen, Nov 15 2019

cp's OEIS Frontend

A231093 Initial members of abundant octuplets, i.e., values of n such that (n, n+2, n+4, n+6, n+8, n+10, n+12, n+14) are all abundant numbers.

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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.

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

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

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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.

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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.

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A303741 Numbers k such that A(k+1) = A(k) + 2, where A() = A005101() are the abundant numbers.

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A306952 Lesser member of twin weird numbers: weird numbers n (A006037) such that n+2 is also weird.

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