A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n).

63, 18, 45, 50, 75, 66, 63, 102, 75, 50, 165, 198, 147, 258, 165, 110, 663, 182, 399, 442, 147, 242, 705, 678, 455, 786, 483, 182, 1015, 950, 1023, 988, 363, 506, 637, 1446, 1083, 322, 885, 590, 1155, 1443, 1935, 2118, 627, 770, 3243, 2502, 1407, 2706, 845

Offset

1,1

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: 63, 18, 45, ...

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..2500

Formula

a(n) >= n^2 + 4n + 6. [Charles R Greathouse IV, Dec 13 2011]

Example

a(8) = 102 because the prime divisors of 102 are 2, 3 and 17;
(2 + 8) | (102 + 8) = 110 = 10*11;
(3 + 8) | 110 = 11*10.

Maple

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:

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] <= 1, k++]; k]; Array[a, 50] (* Amiram Eldar, Sep 09 2019 *)

Crossrefs

Cf. A006972, A029591, A202158, A202159.

Sequence in context: A234692 A339458 A084507 * A145585 A033383 A090635

Adjacent sequences: A202154 A202155 A202156 * A202158 A202159 A202160

Keyword

nonn

Author

Michel Lagneau, Dec 13 2011

Status

approved