A078458 - OEIS

A078458

Total number of factors in a factorization of n into Gaussian primes.

0, 2, 1, 4, 2, 3, 1, 6, 2, 4, 1, 5, 2, 3, 3, 8, 2, 4, 1, 6, 2, 3, 1, 7, 4, 4, 3, 5, 2, 5, 1, 10, 2, 4, 3, 6, 2, 3, 3, 8, 2, 4, 1, 5, 4, 3, 1, 9, 2, 6, 3, 6, 2, 5, 3, 7, 2, 4, 1, 7, 2, 3, 3, 12, 4, 4, 1, 6, 2, 5, 1, 8, 2, 4, 5, 5, 2, 5, 1, 10, 4, 4, 1, 6, 4, 3, 3, 7, 2, 6, 3, 5, 2, 3, 3, 11, 2, 4, 3, 8, 2, 5, 1, 8

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n)+1 is also the total number of factors in a factorization of n+n*i into Gaussian primes. - Jason Kimberley, Dec 17 2011

Record high values are at a(2^k) = 2*k for k = 0, 1, 2, ... . - Bill McEachen, Oct 11 2022

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

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

Eric Weisstein's World of Mathematics, Gaussian Prime.

Index entries for Gaussian integers and primes.

FORMULA

Fully additive with a(p)=2 if p=2 or p mod 4=1 and a(p)=1 if p mod 4=3. - Vladeta Jovovic, Jan 20 2003

a(n) depends on the number of primes of the forms 4k+1 (A083025) and 4k-1 (A065339) and on the highest power of 2 dividing n (A007814): a(n) = 2*A007814(n) + 2*A083025(n) + A065339(n). - T. D. Noe, Jul 14 2003

EXAMPLE

2 = (1+i)*(1-i), so a(2) = 2; 9 = 3*3, so a(9) = 2.

a(1006655265000) = a(2^3*3^2*5^4*7^5*11^3) = 3*a(2)+2*a(3)+4*a(5)+5*a(7)+3*a(11) = 3*2+2*1+4*2+5*1+3*1 = 24. - Vladeta Jovovic, Jan 20 2003

MATHEMATICA

Join[{0}, Table[f = FactorInteger[n, GaussianIntegers -> True]; cnt = Total[Transpose[f][[2]]]; If[MemberQ[{-1, I, -I}, f[[1, 1]]], cnt--]; cnt, {n, 2, 100}]] (* T. D. Noe, Mar 31 2014 *)

a[n_]:=PrimeOmega[n, GaussianIntegers -> True]; Array[a, 104] (* Stefano Spezia, Sep 29 2024 *)

PROG

(PARI) a(n)=my(f=factor(n)); sum(i=1, #f~, if(f[i, 1]%4==3, 1, 2)*f[i, 2]) \\ Charles R Greathouse IV, Mar 31 2014

CROSSREFS

Cf. A078908-A078911, A007814, A065339, A083025.

Cf. A239626, A239627 (including units).

Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), this sequence ("Omega", A001222), A318608 ("mu", A008683).

Equivalent in the ring of Eisenstein integers: A319444.

Sequence in context: A339046 A265911 A363893 * A033317 A183200 A326732

Adjacent sequences: A078455 A078456 A078457 * A078459 A078460 A078461

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane, Jan 11 2003

EXTENSIONS

More terms from Vladeta Jovovic, Jan 12 2003

STATUS

approved