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

8855, 11590, 27885, 122360, 16555, 10290, 6545, 61642, 71799, 65195, 14245, 142788, 63635, 580930, 39585, 21098, 69003, 258482, 59885, 378952, 8715, 266090, 133285, 690501, 27335, 704790, 1017423, 299222, 187891, 771650, 293405, 1638598, 282315, 553610, 227205

Offset: 1

Author

Michel Lagneau, Dec 13 2011

Keywords

• nonn

Comments

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).

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

In this sequence, the majority of terms are not squarefree.

Examples

			a(3) = 27885 because the prime divisors of 27885 are 3, 5, 11, 13  =>
(3 + 3)| (27885 + 3) = 27888 = 6*4648;
(5 + 3) | 27888 = 8*3486;
(11 + 3) | 27888 = 14*1992;
(13 + 3) | 27888 = 16*1743.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..450

Crossrefs

Cf. A006972, A029591, A202157, A202158.

Programs

Maple

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:

Mathematica

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 *)