A316099 Abundant numbers that differ from the next abundant number by 6.

12, 24, 30, 42, 48, 60, 72, 90, 114, 120, 126, 132, 144, 150, 162, 168, 180, 186, 210, 228, 234, 240, 246, 252, 264, 282, 288, 294, 312, 324, 330, 342, 354, 372, 384, 402, 408, 420, 426, 432, 450, 468, 480, 492, 504, 510, 522, 534, 552, 564, 582, 588, 594, 600

Offset: 1

Author

Muniru A Asiru, Jun 25 2018

Keywords

• nonn

Comments

From Amiram Eldar, Sep 02 2022: (Start)

All the terms are even, since all the multiples of 6 that are larger than 6 are abundant numbers.

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 8, 85, 865, 8716, 87668, 875528, 8761027, 87606693, 875947187, ... . Apparently, the asymptotic density of this sequence exists and equals 0.087... . (End)

Examples

			12 is abundant, 13, 14, 15, 16 and 17 are deficient, 18 is abundant.
24 is abundant, 25, 26, 27, 28 and 29 are deficient, 30 is abundant.
		

Links

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

Crossrefs

Subsequence of A005101.

Cf. A316097, A343302.

Cf. A231626 (which has many common terms when 1 is subtracted).

Programs

GAP

A:=Filtered([1..800],n->Sigma(n)>2*n);;  a:=List(Filtered([1..Length(A)-1],i->A[i+1]-A[i]=6),j->A[j]);

Maple

with(numtheory):  A:=select(n->sigma(n)>2*n,[$1..800]): a:=seq(A[i],i in select(n->A[n+1]-A[n]=6,[$1..nops(A)-1]));

Mathematica

q[n_] := DivisorSigma[1, n] > 2 n; Select[Range[600], q[#] && SelectFirst[# + Range[6], q] == # + 6 &] (* Giovanni Resta, Jul 01 2018 *)

PARI

list(lim) = {my(k = 1, k2); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 6, print1(k1, ", ")); k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Formula

a(n) = A005101(A316097(n)). - Amiram Eldar, Mar 01 2025