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

%I A316099 #24 Mar 01 2025 03:20:24
%S A316099 12,24,30,42,48,60,72,90,114,120,126,132,144,150,162,168,180,186,210,
%T A316099 228,234,240,246,252,264,282,288,294,312,324,330,342,354,372,384,402,
%U A316099 408,420,426,432,450,468,480,492,504,510,522,534,552,564,582,588,594,600
%N A316099 Abundant numbers that differ from the next abundant number by 6.
%C A316099 From _Amiram Eldar_, Sep 02 2022: (Start)
%C A316099 All the terms are even, since all the multiples of 6 that are larger than 6 are abundant numbers.
%C A316099 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)
%H A316099 Muniru A Asiru, <a href="/A316099/b316099.txt">Table of n, a(n) for n = 1..10000</a>
%F A316099 a(n) = A005101(A316097(n)). - _Amiram Eldar_, Mar 01 2025
%e A316099 12 is abundant, 13, 14, 15, 16 and 17 are deficient, 18 is abundant.
%e A316099 24 is abundant, 25, 26, 27, 28 and 29 are deficient, 30 is abundant.
%p A316099 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]));
%t A316099 q[n_] := DivisorSigma[1, n] > 2 n; Select[Range[600], q[#] && SelectFirst[# + Range[6], q] == # + 6 &] (* _Giovanni Resta_, Jul 01 2018 *)
%o A316099 (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]);
%o A316099 (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
%Y A316099 Subsequence of A005101.
%Y A316099 Cf. A316097, A343302.
%Y A316099 Cf. A231626 (which has many common terms when 1 is subtracted).
%K A316099 nonn
%O A316099 1,1
%A A316099 _Muniru A Asiru_, Jun 25 2018