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

Offset: 1

Author

N. J. A. Sloane, Jan 11 2003

Keywords

• nonn
• easy

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

Examples

			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
		

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.

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.

Programs

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 *)

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

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

Extensions

More terms from Vladeta Jovovic, Jan 12 2003