A274443 Least composite squarefree number k such that (p-n) | (k-1) for all primes p dividing n.

561, 21, 85, 15, 21, 35, 33, 21, 65, 91, 57, 91, 133, 55, 161, 91, 57, 133, 33, 253, 65, 91, 145, 115, 217, 451, 161, 703, 253, 551, 561, 253, 481, 217, 129, 451, 301, 1081, 161, 1189, 145, 989, 217, 235, 481, 703, 649, 329, 265, 1081, 1121, 1219, 145, 1037, 721

Offset: 1

Author

Paolo P. Lava, Jun 23 2016

Keywords

• nonn
• easy

Examples

			Prime factors of 561 are 3, 11 and 17: (561 - 1) / (3 - 1) = 560 / 2 = 280, (561 - 1) / (11 - 1) = 560 / 10 = 56 and (561 - 1) / (17 - 1) = 560 / 16 = 35.
Prime factors of 21 are 3 and 7: (21 - 1) / (3 - 2) = 20 / 1 = 20, (21 - 1) / (7 - 2) = 20 / 5 = 4.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274444, A274445, A274446.

Programs

Maple

with(numtheory); P:=proc(q) local d,k,n,ok,p;
for k from 1 to q do for n from 2 to q do
if not isprime(n) and issqrfree(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][1]=k then ok:=0; break; else
if not type((n-1)/(p[d][1]-k),integer) then ok:=0; break; fi; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);

Mathematica

t = Select[Range@2000, SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, If[# == 0, False, Divisible[k - 1, #]] &[# - n] &]]], {n, 55}] (* Michael De Vlieger, Jun 24 2016, Version 10 *)