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

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

1,3

Comments

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

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Peter Luschny, Generalized Bernoulli numbers.

Example

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

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 *)

Prog

(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)

Crossrefs

Cf. A226039, A226040, A225481.

Sequence in context: A221178 A080389 A359755 * A181062 A338031 A091199

Adjacent sequences: A226035 A226036 A226037 * A226039 A226040 A226041

Keyword

nonn

Author

Peter Luschny, May 27 2013

Status

approved