A231086 - OEIS

%I #49 Mar 01 2025 03:19:49

%S 18,40,54,70,78,88,100,102,112,138,160,174,196,198,208,220,222,258,

%T 270,280,304,306,318,340,348,350,352,364,366,378,390,400,414,438,448,

%U 460,462,474,490,498,520,532,544,550,558,570,580,606,616,618,640,642,648

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

%C 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

%C From _Amiram Eldar_, May 30 2023: (Start)

%C The least odd term is larger than 10^11.

%C 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)

%H Amiram Eldar, <a href="/A231086/b231086.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from Shyam Sunder Gupta)

%F a(n) = A005101(A303741(n)). - _Amiram Eldar_, Mar 01 2025

%e 18, 20 are abundant, thus the smaller number is listed.

%p 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

%t 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

%t 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 *)

%o (PARI) is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 \\ _Charles R Greathouse IV_, Feb 21 2017

%o (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

%Y Cf. A005101, A096399, A108926, A231088, A231089, A231090, A231092, A231093, A303741.

%K nonn,changed

%O 1,1

%A _Shyam Sunder Gupta_, Nov 03 2013