A057368 Number of Gaussian primes (in the first half-quadrant; i.e., 0 to 45 degrees) with real part = n.

1, 1, 2, 1, 2, 2, 2, 3, 1, 4, 3, 1, 4, 3, 3, 3, 4, 3, 5, 6, 2, 4, 6, 3, 7, 6, 4, 4, 4, 4, 8, 6, 5, 6, 8, 5, 6, 7, 3, 9, 5, 5, 9, 8, 7, 9, 7, 7, 10, 8, 6, 9, 10, 5, 8, 8, 6, 10, 12, 8, 11, 10, 6, 9, 15, 5, 11, 11, 4, 11, 14, 6, 12, 10, 12, 11, 9, 8, 12, 19, 10, 15, 10, 8, 19, 11, 8, 11, 14, 15, 13

Offset: 1

Author

Robert G. Wilson v, Sep 22 2000

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0. - Franklin T. Adams-Watters, May 05 2006

The graph of this sequence inspires the following conjecture: A > a(n)/pi(n) > B, where A and B are constants and pi(n) is the prime counting function (A000720). - T. D. Noe, Feb 26 2007

Stronger conjecture: Let pi(n) be the prime counting function (A000720). Then pi(n) >= a(n) >= pi(n)/5 for n>1, with the following equalities: pi(2)=a(2), pi(3)=a(3), pi(10)=a(10) and a(12)=pi(12)/5. - T. D. Noe, Feb 26 2007

References

• Mark A. Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, New York, 1999, page 269.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Index entries for Gaussian integers and primes

Crossrefs

Cf. A055683 and A057352.

Cf. A069004.

Programs

Mathematica

Do[ c=0; Do[ If[ PrimeQ[ j + k*I, GaussianIntegers -> True ], c++ ], {j, n, n}, {k, 0, j} ]; Print[ c ], {n, 1, 75} ]

Formula

a(n) = A069004(n) + 1 if n is 1 or a prime = 3 (mod 4), A069004(n) otherwise. - Franklin T. Adams-Watters, May 05 2006

a(n) = O(n/log(n)). - Thomas Ordowski, Mar 06 2017

Extensions

More terms from Franklin T. Adams-Watters, May 05 2006