|
| |
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
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
|
|
|
|
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
|
(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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
