A222030 Primes and quarter-squares.

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 16, 17, 19, 20, 23, 25, 29, 30, 31, 36, 37, 41, 42, 43, 47, 49, 53, 56, 59, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 90, 97, 100, 101, 103, 107, 109, 110, 113, 121, 127, 131, 132, 137, 139, 144, 149, 151, 156, 157, 163, 167, 169

Offset: 0

Author

Omar E. Pol, Feb 05 2013

Keywords

• nonn
• easy

Comments

Union of A002620 and A000040.

It appears that there is always a prime between two consecutive quarter squares, if n >= 2. Therefore in a square spiral, or zig-zag, whose vertices are the quarter-squares, it appears that there is always a prime between two consecutive vertices, if n >= 2.

Apparently the above comment is equivalent to the Oppermann's conjecture. - Omar E. Pol, Oct 26 2013

Union of A000040 and A000290 and A002378. - Omar E. Pol, Oct 28 2013

Links

• Wikipedia, Oppermann's conjecture

Crossrefs

Cf. A000040, A002620, A000290, A014085, A220492, A220506, A220508, A220516.

Programs

Mathematica

mx = 13; Union[Prime[Range[PrimePi[mx^2]]], Floor[Range[2*mx]^2/4]] (* Alonso del Arte, Mar 03 2013 *)

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Mar 04 2013