A077767 Number of primes of form 4k+3 between n^2 and (n+1)^2.

1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 4, 5, 3, 4, 4, 4, 3, 5, 4, 4, 5, 5, 4, 4, 5, 5, 4, 8, 8, 5, 4, 6, 5, 6, 7, 5, 5, 7, 5, 7, 7, 7, 6, 8, 4, 5, 11, 5, 9, 8, 6, 11, 7, 7, 7, 7, 8, 10, 5, 12, 10, 5, 9, 10, 7, 13, 8, 8, 11, 5, 10, 9, 13, 9, 6, 9, 12, 7, 7, 11, 10, 9, 12, 11, 10, 10

Offset: 1

Author

T. D. Noe, Nov 20 2002

Keywords

• nonn

Comments

Related to Legendre's conjecture that there is always a prime between two consecutive squares.

Examples

			a(8)=3 because primes 67, 71 and 79 are between squares 64 and 81
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A002145, A014085, A077766.

Programs

Mathematica

maxN=100; a=Table[0, {maxN}]; maxP=PrimePi[(maxN+1)^2]; For[i=1, i<=maxP, i++, p=Prime[i]; If[Mod[p, 4]==3, j=Floor[Sqrt[p]]; a[[j]]++ ]]; a
p3[{a_,b_}]:=Module[{p=Prime[Range[PrimePi[a]+1,PrimePi[b]]]},Count[p,?(Mod[#,4]==3&)]]; p3/@Partition[Range[100]^2,2,1] (* _Harvey P. Dale, Feb 20 2020 *)