A096536 Numbers k such that k, k+1, k+2 are all abundant.

171078830, 268005374, 321893648, 336038624, 487389824, 600350750, 663249950, 668645054, 938109248, 1053424448, 1079741024, 1102433408, 1139364224, 1148927624, 1267293950, 1275861950, 1310259950, 1344330350, 1352253824

Offset: 1

Author

John L. Drost, Aug 13 2004

Keywords

• nonn

Comments

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).

From Amiram Eldar, Oct 02 2022: (Start)

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.

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)

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..22583 (terms below 10^12; terms 1..1000 from Donovan Johnson)
• Carlos Rivera, Puzzle 878. Consecutive abundant integers, The Prime Puzzles & Problems Connection.
• Carlos Rivera, Puzzle 880. Consecutive odd abundant integers, The Prime Puzzles & Problems Connection.

Crossrefs

Subsequence of A005101 and A096399.

Cf. A119240.

Programs

PARI

isab(x) = sigma(x) > 2*x; \\ A005101
isok(k) = isab(k) && isab(k+1) && isab(k+2); \\ Michel Marcus, Nov 19 2022

Extensions

a(15)-a(19) from Donovan Johnson, Dec 29 2008