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A078458
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Total number of factors in a factorization of n into Gaussian primes.
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15
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Jan 11 2003
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EXTENSIONS
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More terms from Vladeta Jovovic, Jan 12 2003
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STATUS
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approved
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