A226038 Numbers k such that there are no primes p which divide k+1 and p-1 does not divide k.

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 22, 24, 26, 28, 30, 31, 36, 40, 42, 44, 46, 48, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 88, 96, 100, 102, 106, 108, 112, 120, 124, 126, 127, 130, 136, 138, 148, 150, 156, 162, 166, 168, 172, 178, 180, 190, 192, 196, 198

Offset: 1

Author

Peter Luschny, May 27 2013

Keywords

• nonn
• changed

Comments

These are the numbers which satisfy the weak Clausen condition but not the Clausen condition.

Examples

			A counterexample is n = 14. 5 divides 15 but 4 does not divide 14.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Peter Luschny, Generalized Bernoulli numbers.

Crossrefs

Cf. A226039, A226040, A225481.

Programs

Maple

s := (p,n) -> ((n+1) mod p = 0) and (n mod (p-1) <> 0);
F := n -> select(p -> s(p,n), select('isprime', [$2..n]));
A226038_list := n -> select(k -> [] = F(k), [$0..n]);
A226038_list(200);

Mathematica

s[p_, n_] := Mod[n+1, p] == 0 && Mod[n, p-1] != 0; f[n_] := Select[ Select[ Range[n], PrimeQ], s[#, n] &]; A226038 = Select[ Range[0, 200], f[#] == {} &] (* Jean-François Alcover, Jul 29 2013, after Maple *)
Join[{0}, Select[Range[200], And @@ Divisible[#, FactorInteger[# + 1][[All, 1]] - 1] &]] (* Ivan Neretin, Aug 04 2016 *)

Sage

def F(n): return filter(lambda p: ((n+1) % p == 0) and (n % (p-1) != 0), primes(n))
def A226038_list(n): return list(filter(lambda n: not list(F(n)), (0..n)))
A226038_list(200)