A202157 - OEIS

%I #23 Sep 09 2019 04:24:34

%S 63,18,45,50,75,66,63,102,75,50,165,198,147,258,165,110,663,182,399,

%T 442,147,242,705,678,455,786,483,182,1015,950,1023,988,363,506,637,

%U 1446,1083,322,885,590,1155,1443,1935,2118,627,770,3243,2502,1407,2706,845

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

%C 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 If k is squarefree, for n = 1, we obtain Lucas-Carmichael numbers: A006972.

%C In this sequence, the majority of terms are not squarefree: 63, 18, 45, ...

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.

%H Amiram Eldar, <a href="/A202157/b202157.txt">Table of n, a(n) for n = 1..2500</a>

%F a(n) >= n^2 + 4n + 6. [_Charles R Greathouse IV_, Dec 13 2011]

%e a(8) = 102 because the prime divisors of 102 are 2, 3 and 17;

%e (2 + 8) | (102 + 8) = 110 = 10*11;

%e (3 + 8) | 110 = 11*10.

%p 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 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 Cf. A006972, A029591, A202158, A202159.

%K nonn

%O 1,1

%A _Michel Lagneau_, Dec 13 2011