A228382 Abundant numbers that differ from the next abundant number by 3.

942, 945, 1572, 1575, 2202, 2205, 2832, 2835, 3462, 3465, 4092, 4095, 4722, 4725, 5352, 5355, 5772, 5985, 6432, 6435, 6612, 6615, 6822, 6825, 7242, 7245, 7425, 7872, 7875, 8082, 8085, 8412, 8415, 8502, 8505, 8922, 8925, 9132, 9135, 9552, 9555, 9762, 9765

Offset

1,1

Comments

Apparently these numbers come up mostly by pairs m, m+3 with m even; the odd terms being a subsequence of A005231. But this is not always the case (e.g., note the term 7425).

The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 2, 43, 393, 3635, 37599, 374092, 3731903, 37338208, 373256850, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00373... . - Amiram Eldar, May 30 2023

Links

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Formula

a(n) = A005101(A316095(n)). - Amiram Eldar, Mar 01 2025

Example

942 is abundant, 943 and 944 are deficient, 945 is abundant.
945 is abundant, 946 and 947 are deficient, 948 is abundant.

Maple

with(numtheory): select(n->sigma(n)>2*n and sigma(n+1)<2*(n+1) and sigma(n+2)<2*(n+2) and sigma(n+3)>2*(n+3), [$1..12000]); # Muniru A Asiru, Jun 09 2018

Mathematica

With[{abs = Select[Range[10000], DivisorSigma[-1, #] > 2 &]}, abs[[Position[Differences[abs], 3] // Flatten]]] (* Amiram Eldar, May 30 2023 *)
SequencePosition[Table[If[DivisorSigma[1, n]>2n, 1, 0], {n, 10000}], {1, 0, 0, 1}][[;; , 1]] (* Harvey P. Dale, Aug 31 2024 *)

Prog

(PARI) isok(n) = (sigma(n)> 2*n) && (sigma(n+1)< 2*(n+1)) && (sigma(n+2) < 2*(n+2)) && (sigma(n+3) > 2*(n+3)); \\ Michel Marcus, Aug 21 2013
(GAP) a:=Filtered([1..130000], n->Sigma(n)>2*n and Sigma(n+1)<2*(n+1) and Sigma(n+2)<2*(n+2) and Sigma(n+3)>2*(n+3)); # Muniru A Asiru, Jun 09 2018

Crossrefs

Subsequence of A005101.

Cf. A005231, A096399, A316095.

Sequence in context: A052238 A158718 A104302 * A340876 A252295 A252294

Adjacent sequences: A228379 A228380 A228381 * A228383 A228384 A228385

Keyword

nonn,changed

Author

Michel Marcus, Aug 21 2013

Status

approved