A089166 Number of primes between squares of successive odd numbers.

4, 5, 6, 7, 8, 9, 9, 13, 11, 13, 14, 15, 15, 17, 16, 19, 19, 19, 21, 23, 20, 23, 23, 28, 21, 31, 25, 23, 30, 32, 30, 29, 31, 32, 34, 30, 34, 40, 32, 36, 39, 37, 37, 40, 38, 39, 44, 42, 45, 44, 42, 42, 45, 42, 48, 52, 49, 45, 50, 48, 51, 55, 56, 47, 52, 56, 56, 53, 49, 58, 62, 56

Offset: 1

Author

Robert G. Wilson v, Dec 06 2003

Keywords

• nonn

Comments

As the squares of the successive odd numbers are those numbers appearing on the SE spoke of the Ulam spiral, a(n) also gives the number of primes appearing in the n-th square ring around 1 of the Ulam spiral. - Scott R. Shannon, Jan 14 2020

Links

• Didier van der Straten, Jon Perry, Mark Underwood, TR: [PrimeNumbers] pi(x), digest of 3 messages in primeforms Yahoo group, Aug 29 - Aug 30, 2003. [Cached copy]
• Mark Underwood, [PrimeNumbers] pi(x).
• Wikipedia, Ulam Spiral.

Crossrefs

Cf. A014085, A016754 (odd squares).

Programs

Magma

[#[p:p in PrimesInInterval(k^2,(k+2)^2)]:k in [1..150 by 2]]; // Marius A. Burtea, Jan 14 2020

Mathematica

Table[PrimePi[(2n + 1)^2] - PrimePi[(2n - 1)^2], {n, 1, 72}]

PARI

forstep (k=1,130,2,print1(primepi((k+2)^2)-primepi(k^2),", ")) \\ Hugo Pfoertner, Nov 15 2019