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

Author

Michel Marcus, Aug 21 2013

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Subsequence of A005101.

Cf. A005231, A096399, A316095.

Programs

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

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 *)

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

Formula

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