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A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n).

%I A202157 #23 Sep 09 2019 04:24:34
%S A202157 63,18,45,50,75,66,63,102,75,50,165,198,147,258,165,110,663,182,399,
%T A202157 442,147,242,705,678,455,786,483,182,1015,950,1023,988,363,506,637,
%U A202157 1446,1083,322,885,590,1155,1443,1935,2118,627,770,3243,2502,1407,2706,845
%N A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n).
%C A202157 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 A202157 If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.
%C A202157 In this sequence, the majority of terms are not squarefree: 63, 18, 45, ...
%D A202157 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.
%H A202157 Amiram Eldar, <a href="/A202157/b202157.txt">Table of n, a(n) for n = 1..2500</a>
%F A202157 a(n) >= n^2 + 4n + 6. [_Charles R Greathouse IV_, Dec 13 2011]
%e A202157 a(8) = 102 because the prime divisors of 102 are 2, 3 and 17;
%e A202157 (2 + 8) | (102 + 8) = 110 = 10*11;
%e A202157 (3 + 8) | 110 = 11*10.
%p A202157 with(numtheory):for n from 1 to 52 do:i:=0:for k from 1 to 5000 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 A202157 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] <= 1, k++]; k]; Array[a, 50] (* _Amiram Eldar_, Sep 09 2019 *)
%Y A202157 Cf. A006972, A029591, A202158, A202159.
%K A202157 nonn
%O A202157 1,1
%A A202157 _Michel Lagneau_, Dec 13 2011