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A078458 Total number of factors in a factorization of n into Gaussian primes. 15
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

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 W. Weisstein, MathWorld: 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 *)

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, A086275 (number of distinct Gaussian primes in the factorization of n).

Cf. A239626, A239627 (including units).

Sequence in context: A153281 A130584 A265911 * A033317 A183200 A305422

Adjacent sequences:  A078455 A078456 A078457 * A078459 A078460 A078461

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jan 11 2003

EXTENSIONS

More terms from Vladeta Jovovic, Jan 12 2003

STATUS

approved

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Last modified April 25 23:48 EDT 2019. Contains 322465 sequences. (Running on oeis4.)