A295585 Numbers k such that Dirichlet's theorem has a purely elementary proof mod k via the Erdős method.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 36, 40, 42, 48, 50, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 150, 156, 168, 180, 210, 240, 270, 300, 330, 390, 420, 630, 840

Offset

1,2

Comments

Moree gives an effective version, see Theorem 1.

Links

Table of n, a(n) for n=1..55.

Paul Erdős, Über die Primzahlen gewisser arithmetischer Reihen, Math. Z. 39 (1935), pp. 473-491. [alternate link]

Martin Klazar, Analytic and Combinatorial Number Theory II (lecture notes). See section 2.3, Erdős's partial proof of Dirichlet's theorem.

P. Moree, Bertrand's postulate for primes in arithmetical progressions, Computers & Mathematics with Applications 26:5 (1993), pp. 35-43.

Formula

Numbers k such that Sum_{p < k, p does not divide k} 1/p < 1.

Example

15 is in the sequence since 1/2 + 1/7 + 1/11 + 1/13 = 1623/2002 < 1.

Prog

(PARI) is(n)=if(n>840, 0, my(s); forprime(p=2, n-1, if(n%p, s+=1/p)); s<1)

Crossrefs

Sequence in context: A225734 A225733 A225732 * A324578 A373847 A193988

Adjacent sequences: A295582 A295583 A295584 * A295586 A295587 A295588

Keyword

nonn,fini,full

Author

Charles R Greathouse IV, Nov 23 2017

Status

approved