A078909 Let r+i*s be the sum, with multiplicity, of the first-quadrant Gaussian primes dividing n; sequence gives s values.

0, 2, 0, 4, 3, 2, 0, 6, 0, 5, 0, 4, 5, 2, 3, 8, 5, 2, 0, 7, 0, 2, 0, 6, 6, 7, 0, 4, 7, 5, 0, 10, 0, 7, 3, 4, 7, 2, 5, 9, 9, 2, 0, 4, 3, 2, 0, 8, 0, 8, 5, 9, 9, 2, 3, 6, 0, 9, 0, 7, 11, 2, 0, 12, 8, 2, 0, 9, 0, 5, 0, 6, 11, 9, 6, 4, 0, 7, 0, 11, 0, 11, 0, 4, 8, 2, 7, 6, 13, 5, 5, 4, 0, 2, 3, 10, 13, 2, 0

Offset

1,2

Comments

A Gaussian integer z = x+iy is in the first quadrant if x > 0, y >= 0. Just one of the 4 associates z, -z, i*z, -i*z is in the first quadrant.

The sequence is fully additive.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Michael Somos, PARI program for finding prime decomposition of Gaussian integers

Index entries for Gaussian integers and primes

Example

5 factors into the product of the primes 1+2*i, 1-2*i, but the first-quadrant associate of 1-2*i is i*(1-2*i) = 2+i, so r+i*s = 1+2*i + 2+i = 3+3*i. Therefore a(5) = 3.

Mathematica

a[n_] := Module[{f = FactorInteger[n, GaussianIntegers->True]}, p = f[[;; , 1]]; e = f[[;; , 2]]; Im[Plus @@ ((If[Abs[#] == 1, 0, #]& /@ p) * e)]]; Array[a, 100] (* Amiram Eldar, Feb 28 2020 *)

Crossrefs

Cf. A078458, A078908, A078910, A078911, A080088, A080089.

Sequence in context: A359214 A265584 A360205 * A067458 A088330 A376252

Adjacent sequences: A078906 A078907 A078908 * A078910 A078911 A078912

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 11 2003

Extensions

More terms and further information from Vladeta Jovovic, Jan 27 2003

Status

approved