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A231086
Initial members of abundant twins, i.e., values of k such that (k, k+2) are both abundant numbers.
12
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
1,1
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)
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Shyam Sunder Gupta)
EXAMPLE
18, 20 are abundant, thus the smaller number is listed.
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 *)
PROG
(PARI) is(n)=sigma(n, -1)>2 && sigma(n+2, -1)>2 \\ Charles R Greathouse IV, Feb 21 2017
(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
KEYWORD
nonn
AUTHOR
Shyam Sunder Gupta, Nov 03 2013
STATUS
approved