A331125 Numbers k such that there is no prime p between k and (9/8)k, exclusive.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 19, 20, 23, 24, 25, 31, 32, 47

Offset: 1

Author

Bernard Schott, Jan 10 2020

Keywords

• nonn
• fini
• full

Comments

In 1932, Robert Hermann Breusch proved that for n >= 48, there is at least one prime p between n and (9/8)n, exclusive (A327802).

The terms of A285586 correspond to numbers k such that there is no prime p between k and (9/8)n, inclusive.

Examples

			Between 16 and (9/8) * 16 = 18, exclusive, there is the prime 17, hence 16 is not a term.
Between 47 and (9/8) * 47 = 52.875, exclusive, 48, 49, 50, 51 and 52 are all composite numbers, hence 47 is a term.
		

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised edition), Penguin Books, 1997, entry 48, p. 106.

Links

• Robert Breusch, Zur Verallgemeinerung des Bertrandschen Postulates, daß zwischen x und 2x stets Primzahlen liegen, Mathematische Zeitschrift (in German), December 1932, Volume 34, Issue 1, pp. 505-526.
• Wikipedia, Robert Breusch

Crossrefs

Cf. A285586, A327802.

Programs

Mathematica

Select[Range[47], Count[Range[# + 1, 9# / 8], ?PrimeQ] == 0 &] (* _Amiram Eldar, Jan 11 2020 *)
Select[Range[1000], PrimePi[#] == PrimePi[9#/8] &] (* Alonso del Arte, Jan 16 2020 *)

Formula

A327802(a(n)) = 0.