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

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

Author

Shyam Sunder Gupta, Nov 03 2013

Keywords

• nonn

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

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

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