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

Offset

1,1

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Yong-Gao Chen, 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.

Example

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

Maple

with(numtheory): P:=proc(n); if sigma(n)>2*n and sigma(n+1)>2*(n+1) then n;
fi; end: seq(P(i), i=1..10^6); # Paolo P. Lava, Jan 08 2018

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 *)

Prog

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

Crossrefs

Cf. A005101, A103289, A103291, A023196.

Sequence in context: A317049 A329525 A331202 * A071132 A228466 A094063

Adjacent sequences: A096396 A096397 A096398 * A096400 A096401 A096402

Keyword

nonn

Author

John L. Drost, Aug 06 2004

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

Status

approved