A202160 a(n) = smallest k having at least five prime divisors d such that (d + n) | (k + n).

588455, 179998, 460317, 6265805, 1236235, 287274, 949025, 1436932, 794871, 2013650, 3797365, 1169688, 3739827, 1587586, 6872565, 7706270, 1529983, 7351242, 2528045, 5247970, 487179, 10920965, 1316497, 121894476, 1404455, 5814874, 12223653, 2260412, 8022531

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) = 460317 because the prime divisors of 460317 are 3, 11, 13, 29, 37  =>
(3 + 3) | (460317 + 3) = 460320 = 6*76720;
(11 + 3) | 460320 = 14*32880;
(13 + 3) | 460320 = 16*28770;
(29+3)  |  460320 = 32*14385;
(37+3) | 460320 = 40*11508.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..125

Crossrefs

Cf. A006972, A029591, A202157, A202158, A202159.

Programs

Maple

with(numtheory):for n from 1 to 23 do:i:=0:for k from 1 to 10^8 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>4 then i:=1: printf ( "%d %d \n",n,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] <= 4, k++]; k]; Array[a, 30] (* Amiram Eldar, Sep 09 2019 *)