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A065338 a(1) = 1, a(p) = p mod 4 for p prime and a(u * v) = a(u) * a(v) for u, v > 0. 18
1, 2, 3, 4, 1, 6, 3, 8, 9, 2, 3, 12, 1, 6, 3, 16, 1, 18, 3, 4, 9, 6, 3, 24, 1, 2, 27, 12, 1, 6, 3, 32, 9, 2, 3, 36, 1, 6, 3, 8, 1, 18, 3, 12, 9, 6, 3, 48, 9, 2, 3, 4, 1, 54, 3, 24, 9, 2, 3, 12, 1, 6, 27, 64, 1, 18, 3, 4, 9, 6, 3, 72, 1, 2, 3, 12, 9, 6, 3, 16, 81, 2, 3, 36, 1, 6, 3, 24, 1, 18, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = (2^A007814(n)) * (3^A065339(n)). a(n) <= n. a(a(n)) = a(n). a(x) = x iff x = 2^i * 3^j for i, j >= 0. a(A003586(n)) = A003586(n). a(A065331(n)) = A065331(n).

a(A004613(n)) = 1; A065333(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2010

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = if n = 1 then 1 else (A020639(n) mod 4) * n / A020639(n).

Dirichlet g.f.: 1/(1-2^(-s+1)) * prod(prime p=4k+1, 1/(1-p^(-s)) * prod(prime p=4k+3, 1/(1-3*p^(-s)). - Ralf Stephan, Mar 28 2015

EXAMPLE

a(120) = a(2*2*2*3*5) = a(2)*a(2)*a(2)*a(3)*a(5) = 2*2*2*3*1 = 24.

a(150) = a(2*3*5*5) = a(2)*a(3)*a(5)*a(5) = 2*3*1*1 = 6.

a(210) = a(2*3*5*7) = a(2)*a(3)*a(5)*a(7) = 2*3*1*3 = 18.

MATHEMATICA

a[1] = 1; a[n_] := a[n] = Mod[p = FactorInteger[n][[1, 1]], 4]*a[n/p]; Table[ a[n], {n, 1, 100} ] (* Jean-Fran├žois Alcover, Jan 20 2012 *)

PROG

(Haskell)

a065338 1 = 1

a065338 n = (spf `mod` 4) * a065338 (n `div` spf) where spf = a020639 n

-- Reinhard Zumkeller, Nov 18 2011

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, (f[i, 1]%4)^f[i, 2]) \\ Charles R Greathouse IV, Feb 07 2017

CROSSREFS

Cf. A039702, A000040, A003586, A007814, A065339, A065331.

Sequence in context: A082119 A129708 A071518 * A316272 A294649 A001438

Adjacent sequences:  A065335 A065336 A065337 * A065339 A065340 A065341

KEYWORD

mult,nice,nonn

AUTHOR

Reinhard Zumkeller, Oct 29 2001

STATUS

approved

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Last modified November 15 13:56 EST 2019. Contains 329149 sequences. (Running on oeis4.)