A096399 -id:A096399 - cp's OEIS frontend

A283418 Numbers n such that n and n+1 are primitive abundant.

82004, 158235, 326864, 442035, 516704, 1102724, 1606275, 2151435, 2697435, 2912084, 2921535, 2979675, 3002804, 3241755, 3647475, 4322835, 5801984, 5905844, 6069195, 7251075, 7387604, 7553924, 8272124, 8788724, 9292724, 9909584

Offset: 1

Emmanuel Vantieghem, May 02 2017

nonn

Intersection of A091191 and -1 + A091191.

			82004 is in the sequence because it is abundant (sum divisors = 164640, > 2*82004) and 82005 is also abundant (sum divisors = 165888, > 2*82005).

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A091191, A005105, A096399.

fQ[m_] := DivisorSigma[1, m] > 2 m;
gQ[m_] := fQ[m] && Union[fQ /@ Rest[Most[Divisors[m]]]] == {False};
V = Select[Range[10^7], gQ]; Intersection[V, V - 1]

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

Shyam Sunder Gupta, Nov 03 2013

nonn

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)

			18, 20 are abundant, thus the smaller 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, A231088, A231089, A231090, A231092, A231093, A303741.

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

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

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 *)

is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

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

A318167 Numbers k such that both k and k+1 are bi-unitary abundant numbers.

Original entry on oeis.org

21735, 21944, 43064, 49664, 58695, 76544, 106784, 135135, 144584, 160544, 188055, 209055, 227744, 256095, 262184, 300104, 345344, 348704, 382304, 387584, 407295, 409184, 414855, 437535, 498015, 520695, 560384, 567944, 611415, 679455, 687015, 705375, 709695

Offset: 1

Amiram Eldar, Aug 20 2018

nonn

The bi-unitary version of A096399.

			21735 is in the sequence since both 21735 and 21736 are bi-unitary abundant numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A096399 (analog for sigma), A188999 (bi-unitary sigma).

Cf. A292982 (bi-unitary abundant), A293186 (odd bi-unitary abundant).

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 2 n; seq={}; n=1; While[Length[seq]<32,If[bAbundantQ[n] && bAbundantQ [n+1],AppendTo[seq,n]];n++];seq

a188999(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
isok(n) = (a188999(n) > 2*n) && (a188999(n+1) > 2*(n+1)); \\ Michel Marcus, Aug 21 2018

A327635 Numbers k such that both k and k+1 are infinitary abundant numbers (A129656).

Original entry on oeis.org

21735, 21944, 43064, 58695, 188055, 262184, 414855, 520695, 567944, 611415, 687015, 764504, 792855, 809864, 812889, 833624, 874664, 911624, 945944, 976184, 991304, 1019655, 1026375, 1065015, 1073709, 1157624, 1201095, 1218944, 1248344, 1254015, 1272375, 1272704

Offset: 1

Amiram Eldar, Sep 20 2019

nonn

The least k such that k, k+1 and k+2 are all infinitary abundant numbers is a(75976) = 2666847104.

			21735 is in the sequence since both 21735 and 21736 are infinitary abundant: isigma(21735) = 46080 > 2 * 21735, and isigma(21736) = 50400 > 2 * 21736 (isigma is the sum of infinitary divisors, A049417).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049417, A127666, A129656.

Cf. A096399, A292704, A318167.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); abQ[n_] := isigma[n] > 2n; s={}; ab1 = 0; Do[ab2 = abQ[n]; If[ab1 && ab2, AppendTo[s, n-1]]; ab1 = ab2, {n, 2, 10^5}]; s

A096536 Numbers k such that k, k+1, k+2 are all abundant.

Original entry on oeis.org

171078830, 268005374, 321893648, 336038624, 487389824, 600350750, 663249950, 668645054, 938109248, 1053424448, 1079741024, 1102433408, 1139364224, 1148927624, 1267293950, 1275861950, 1310259950, 1344330350, 1352253824

Offset: 1

John L. Drost, Aug 13 2004

nonn

The entries shown are all even, the first odd k would have to have sigma(k*(k+2)) > 4k*(k+2) so k > 10^19 (cf. A119240).
From Amiram Eldar, Oct 02 2022: (Start)
The least term that is == 1 (mod 3) is a(1292) = 55959128224, and the least term that is divisible by 3 is a(1590) = 68972878974.
The numbers of terms not exceeding 10^k, for k = 9, 10, ..., are 9, 226, 2298, 22583, ... . Apparently, the asymptotic density of this sequence exists and equals 2.2...*10^(-8). (End)

			For 171078830 = 2*5*13*23*29*1973, sigma(n)/n = 2.09355, for 171078831 = 3^3*7*11*19*61*71, sigma(n)/n = 2.00396 and for 171078832 = 2^4*31*344917, sigma(n)/n = 2.00000579.

Amiram Eldar, Table of n, a(n) for n = 1..22583 (terms below 10^12; terms 1..1000 from Donovan Johnson)
Carlos Rivera, Puzzle 878. Consecutive abundant integers, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 880. Consecutive odd abundant integers, The Prime Puzzles & Problems Connection.

Subsequence of A005101 and A096399.

Cf. A119240.

isab(x) = sigma(x) > 2*x; \\ A005101
isok(k) = isab(k) && isab(k+1) && isab(k+2); \\ Michel Marcus, Nov 19 2022

a(15)-a(19) from Donovan Johnson, Dec 29 2008

A331412 Unitary abundant numbers k such that k + 1 is also unitary abundant.

Original entry on oeis.org

8857357509, 10783550414, 15197873690, 23620285689, 25537083494, 34736070369, 60326914934, 64139567205, 73969772954, 75776483145, 77509981185, 83968675790, 93092467754, 100012014465, 112236593469, 113606741534, 116519300534, 118905484334, 132584489114, 134889106065

Offset: 1

Amiram Eldar and Giovanni Resta, Jan 18 2020

nonn

Apparently most of the terms are squarefree. Up to 10^13 there are 1150 terms, for only 17 terms k either k or k + 1 is nonsquarefree, and there are no terms k such that both k and k + 1 are nonsquarefree. The first nonsquarefree term is a(32) = 285491549265.

			8857357509 is a term since usigma(8857357509) = 17766604800 > 2 * 8857357509, and usigma(8857357510) = 17851083264 > 2 * 8857357510, where usigma is the sum of unitary divisors function (A034448).

Giovanni Resta, Table of n, a(n) for n = 1..1150 (terms below 10^13)

Cf. A034448, A034683, A292704.

Analogous sequences: A096399 (regular abundant), A283418 (primitive), A318167 (bi-unitary), A327635 (infinitary), A327942 (nonunitary).

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

A327942 Numbers k such that both k and k+1 are nonunitary abundant numbers (A064597).

Original entry on oeis.org

165375, 893024, 1047375, 1576575, 2282175, 2304224, 2858624, 3614624, 4068224, 4096575, 4597424, 4975424, 6591375, 7574175, 8555624, 9511424, 10446975, 10749375, 10872224, 11477024, 12535424, 13773375, 13946624, 14277375, 15926624, 16041375, 16505775, 16769024

Offset: 1

Amiram Eldar, Sep 30 2019

nonn

			165375 is in the sequence since both 165375 and 165376 are nonunitary abundant: nusigma(165375) = 179280 > 165375, and nusigma(165376) = 183600 > 165376 (nusigma is the sum of nonunitary divisors, A048146).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048146, A064597, A094889, A307823.

Cf. A096399, A292704, A318167, A327635.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); nuabQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (1 + Power @@@ FactorInteger[n]) > n; s = {}; q1 = False; Do[q2 = nuabQ[n]; If[q1 && q2, AppendTo[s, n - 1]]; q1 = q2, {n, 2, 10^7}]; s

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A283418 Numbers n such that n and n+1 are primitive abundant.

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

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A318167 Numbers k such that both k and k+1 are bi-unitary abundant numbers.

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A327635 Numbers k such that both k and k+1 are infinitary abundant numbers (A129656).

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A096536 Numbers k such that k, k+1, k+2 are all abundant.

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A331412 Unitary abundant numbers k such that k + 1 is also unitary abundant.

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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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