A055683 -id:A055683 - cp's OEIS frontend

A057352 Number of Gaussian primes (in the first half-quadrant; i.e., 0 to 45 degrees) with real part <= n.

1, 2, 4, 5, 7, 9, 11, 14, 15, 19, 22, 23, 27, 30, 33, 36, 40, 43, 48, 54, 56, 60, 66, 69, 76, 82, 86, 90, 94, 98, 106, 112, 117, 123, 131, 136, 142, 149, 152, 161, 166, 171, 180, 188, 195, 204, 211, 218, 228, 236

Offset: 1

Robert G. Wilson v, Sep 22 2000

nonn

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

Cf. A055683.

c = 0; Do[ Do[ If[ PrimeQ[ j + k*I, GaussianIntegers -> True ], c++ ], {j, n, n}, {k, 0, j} ]; Print[ c ], {n, 1, 50} ]

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

Original entry on oeis.org

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

Robert G. Wilson v, Sep 22 2000

nonn

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

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

Cf. A055683 and A057352.

Cf. A069004.

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

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

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

A186903 Number of Gaussian primes (in the first half quadrant; i.e., 0 to 45 degrees) with real part <= 2^n.

Original entry on oeis.org

1, 2, 5, 14, 36, 112, 354, 1193, 4099, 14374, 51265, 185143, 675016, 2481768, 9183221, 34179072, 127826852

Offset: 0

Robert G. Wilson v, Mar 02 2011

nonn

Cf. A055683.

c = 1; lst = {}; Do[ Do[ If[ PrimeQ[j + k*I, GaussianIntegers -> True], c++], {j, 2^(n - 1) + 1, 2^n}, {k, 0, j}]; AppendTo[lst, c]; Print@ c, {n, 16}]

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A057352 Number of Gaussian primes (in the first half-quadrant; i.e., 0 to 45 degrees) with real part <= n.

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

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Mathematica

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A186903 Number of Gaussian primes (in the first half quadrant; i.e., 0 to 45 degrees) with real part <= 2^n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica