A096536 - OEIS

%I #36 Nov 19 2022 04:32:11

%S 171078830,268005374,321893648,336038624,487389824,600350750,

%T 663249950,668645054,938109248,1053424448,1079741024,1102433408,

%U 1139364224,1148927624,1267293950,1275861950,1310259950,1344330350,1352253824

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

%C 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).

%C From _Amiram Eldar_, Oct 02 2022: (Start)

%C 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.

%C 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)

%H Amiram Eldar, <a href="/A096536/b096536.txt">Table of n, a(n) for n = 1..22583</a> (terms below 10^12; terms 1..1000 from Donovan Johnson)

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_878.htm">Puzzle 878. Consecutive abundant integers</a>, The Prime Puzzles & Problems Connection.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_880.htm">Puzzle 880. Consecutive odd abundant integers</a>, The Prime Puzzles & Problems Connection.

%e 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.

%o (PARI) isab(x) = sigma(x) > 2*x; \\ A005101

%o isok(k) = isab(k) && isab(k+1) && isab(k+2); \\ _Michel Marcus_, Nov 19 2022

%Y Subsequence of A005101 and A096399.

%Y Cf. A119240.

%K nonn

%O 1,1

%A _John L. Drost_, Aug 13 2004

%E a(15)-a(19) from _Donovan Johnson_, Dec 29 2008