A202160 - OEIS

%I #25 Sep 09 2019 04:24:22

%S 588455,179998,460317,6265805,1236235,287274,949025,1436932,794871,

%T 2013650,3797365,1169688,3739827,1587586,6872565,7706270,1529983,

%U 7351242,2528045,5247970,487179,10920965,1316497,121894476,1404455,5814874,12223653,2260412,8022531

%N a(n) = smallest k having at least five 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.

%H Amiram Eldar, <a href="/A202160/b202160.txt">Table of n, a(n) for n = 1..125</a>

%e a(3) = 460317 because the prime divisors of 460317 are 3, 11, 13, 29, 37  =>

%e (3 + 3) | (460317 + 3) = 460320 = 6*76720;

%e (11 + 3) | 460320 = 14*32880;

%e (13 + 3) | 460320 = 16*28770;

%e (29+3)  |  460320 = 32*14385;

%e (37+3) | 460320 = 40*11508.

%p 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:

%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] <= 4, k++]; k]; Array[a, 30] (* _Amiram Eldar_, Sep 09 2019 *)

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

%K nonn

%O 1,1

%A _Michel Lagneau_, Dec 13 2011