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

399, 598, 165, 1886, 715, 2370, 273, 532, 231, 935, 3445, 828, 1547, 2821, 1105, 3710, 12903, 4182, 6669, 4732, 2475, 4466, 2737, 2706, 1595, 5658, 10413, 3542, 7315, 24225, 23769, 22578, 3927, 12818, 1885, 64119, 11063, 20482, 10881, 4370, 52275, 7878, 14645

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) = 165 because the prime divisors of 165 are 3, 5, 11 =>
(3 + 3) | (165 + 3) = 168 = 6*28;
(5 + 3) | 168 = 8*21;
(11 + 3) | 168 = 14*12.
		

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A006972, A029591, A202157, A202159.

Programs

Maple

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:

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] <= 2, k++]; k]; Array[a, 40] (* Amiram Eldar, Sep 09 2019 *)