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A202158 a(n) = smallest k having at least three prime divisors d such that (d + n) | (k + n).

%I A202158 #23 Sep 09 2019 04:24:29
%S A202158 399,598,165,1886,715,2370,273,532,231,935,3445,828,1547,2821,1105,
%T A202158 3710,12903,4182,6669,4732,2475,4466,2737,2706,1595,5658,10413,3542,
%U A202158 7315,24225,23769,22578,3927,12818,1885,64119,11063,20482,10881,4370,52275,7878,14645
%N A202158 a(n) = smallest k having at least three prime divisors d such that (d + n) | (k + n).
%C A202158 The sequence of numbers k composite and squarefree, prime p | k ==> p+n | k+n is given by A029591 (least quasi-Carmichael number of order -n).
%C A202158 If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
%C A202158 In this sequence, the majority of terms are not squarefree.
%D A202158 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.
%H A202158 Amiram Eldar, <a href="/A202158/b202158.txt">Table of n, a(n) for n = 1..500</a>
%e A202158 a(3) = 165 because the prime divisors of 165 are 3, 5, 11 =>
%e A202158 (3 + 3) | (165 + 3) = 168 = 6*28;
%e A202158 (5 + 3) | 168 = 8*21;
%e A202158 (11 + 3) | 168 = 14*12.
%p A202158 with(numtheory):for n from 1 to 45 do:i:=0:for k from 1 to 100000 while(i=0) do:x:=factorset(k):n1:=nops(x):y:=k+n: j:=0:for m from 1 to n1 do:if  n1>=2 and irem(y,x[m]+n)=0 then j:=j+1:else fi:od:if j>2 then i:=1:printf(`%d, `,k):else fi:od:od:
%t A202158 numd[n_, k_] := Module[{p=FactorInteger[k][[;;,1]], c=0}, Do[If[Divisible[n+k, n+p[[i]]], c++], {i,1,Length[p]}]; c]; a[n_]:=Module[{k=1}, While[numd[n, k] <= 2, k++]; k]; Array[a, 40] (* _Amiram Eldar_, Sep 09 2019 *)
%Y A202158 Cf. A006972, A029591, A202157, A202159.
%K A202158 nonn
%O A202158 1,1
%A A202158 _Michel Lagneau_, Dec 13 2011