A096399 Numbers k such that both k and k+1 are abundant.

5775, 5984, 7424, 11024, 21735, 21944, 26144, 27404, 39375, 43064, 49664, 56924, 58695, 61424, 69615, 70784, 76544, 77175, 79695, 81080, 81675, 82004, 84524, 84644, 89775, 91664, 98175, 103455, 104895, 106784, 109395, 111824, 116655, 116864, 120015, 121904, 122264

Offset: 1

Author

John L. Drost, Aug 06 2004

Keywords

• nonn

Comments

Numbers k such that both sigma(k) > 2k and sigma(k+1) > 2*(k+1).

Numbers k such that both k and k+1 are in A005101.

Set difference of sequences A103289 and {2^m-1} for m in A103291.

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 3, 27, 357, 3723, 36640, 365421, 3665799, 36646071, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000366... . - Amiram Eldar, Sep 02 2022

Examples

			sigma(5775) = sigma(3*5*5*7*11) = 11904 > 2*5775.
sigma(5776) = sigma(2*2*2*2*19*19) = 11811 > 2*5776.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Yong-Gao Chen and Hui Lv, On consecutive abundant numbers, arXiv:1603.06176 [math.NT], 2016.
• Paul Erdős, Note on consecutive abundant numbers, J. London Math. Soc., 10 (1935), 128-131.
• Carlos Rivera, Puzzle 878. Consecutive abundant integers, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A000203, A005101, A023196, A103289, A103291, A169822.

Programs

Mathematica

fQ[n_] := DivisorSigma[1, n] > 2 n; Select[ Range@ 117000, fQ[ # ] && fQ[ # + 1] &] (* Robert G. Wilson v, Jun 11 2010 *)
Select[Partition[Select[Range[120000], DivisorSigma[1, #] > 2 # &], 2, 1], Differences@ # == {1} &][[All, 1]] (* Michael De Vlieger, May 20 2017 *)

PARI

for(i=1,1000000,if(sigma(i)>2*i && sigma(i+1)>2*(i+1),print(i))); \\ Max Alekseyev, Jan 28 2005

Formula

a(n) = A005101(A169822(n)). - Amiram Eldar, Mar 01 2025

Extensions

Two further terms from Max Alekseyev, Jan 28 2005

Entry revised by N. J. A. Sloane, Dec 03 2006

Edited by T. D. Noe, Nov 15 2010