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A202159 a(n) = smallest k having at least four prime divisors d such that (d + n) | (k + n).

%I A202159 #20 Sep 09 2019 04:24:25
%S A202159 8855,11590,27885,122360,16555,10290,6545,61642,71799,65195,14245,
%T A202159 142788,63635,580930,39585,21098,69003,258482,59885,378952,8715,
%U A202159 266090,133285,690501,27335,704790,1017423,299222,187891,771650,293405,1638598,282315,553610,227205
%N A202159 a(n) = smallest k having at least four prime divisors d such that (d + n) | (k + n).
%C A202159 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 A202159 If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
%C A202159 In this sequence, the majority of terms are not squarefree.
%H A202159 Amiram Eldar, <a href="/A202159/b202159.txt">Table of n, a(n) for n = 1..450</a>
%e A202159 a(3) = 27885 because the prime divisors of 27885 are 3, 5, 11, 13  =>
%e A202159 (3 + 3)| (27885 + 3) = 27888 = 6*4648;
%e A202159 (5 + 3) | 27888 = 8*3486;
%e A202159 (11 + 3) | 27888 = 14*1992;
%e A202159 (13 + 3) | 27888 = 16*1743.
%p A202159 with(numtheory):for n from 1 to 33 do:i:=0:for k from 1 to 5000000  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>3 then i:=1:printf(`%d, `,k):else fi:od:od:
%t A202159 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] <= 3, k++]; k]; Array[a, 35] (* _Amiram Eldar_, Sep 09 2019 *)
%Y A202159 Cf. A006972, A029591, A202157, A202158.
%K A202159 nonn
%O A202159 1,1
%A A202159 _Michel Lagneau_, Dec 13 2011