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A202157 a(n) = smallest k having at least two prime divisors d such that (d + n) | ( k + n). 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A234692 A339458 A084507 * A145585 A033383 A090635
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 13 2011
STATUS
approved

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Last modified April 12 11:40 EDT 2024. Contains 371635 sequences. (Running on oeis4.)