A076343 -id:A076343 - cp's OEIS frontend

A076340 Real part of the function defined multiplicatively on the complex numbers by 2->(2,0) and p->((floor(p/4)+floor((p mod 4)/2))*4,2-(p mod 4)) for odd primes p.

1, 2, 4, 4, 4, 8, 8, 8, 15, 8, 12, 16, 12, 16, 17, 16, 16, 30, 20, 16, 31, 24, 24, 32, 15, 24, 52, 32, 28, 34, 32, 32, 47, 32, 33, 60, 36, 40, 49, 32, 40, 62, 44, 48, 68, 48, 48, 64, 63, 30, 65, 48, 52, 104, 49, 64, 79, 56, 60, 68, 60, 64, 112, 64, 47, 94, 68, 64, 95, 66, 72

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

a(n)>0 for n<2187=3^7, a(2187)=-5816, A076341(2187)=-20047.

			n=21: 21 = 3*7 = (4-1)*(8-1) = (4,-1)*(8,-1) -> (32-(-1)*(-1),-4+(-8)) = (31,-12), therefore a(21)=31, A076341(21)=-12;
n=35: 35 = 5*7 = (4+1)*(8-1) = (4,1)*(8,-1) -> (32-1*(-1),-4+8) = (33,4), therefore a(35)=33, A076341(35)=4.

Imaginary part = A076341.
Cf. A076342, A076343, A076345, A076347, A076349.
Cf. A000040, A001358, A000961, A005117, A000290.

b[n_] := If[n == 1, 1, Product[{p, e} = pe; If[p == 2, 2, ((Floor[p/4] + Floor[Mod[p, 4]/2])*4 + (2 - Mod[p, 4]) I)]^e, {pe, FactorInteger[n]}]];
a[n_] := Re[b[n]];
Array[a, 100] (* Jean-François Alcover, Dec 12 2021 *)

a(A000040(n)) = A076342(n).
a(A001358(n)) = A076343(n).
a(A000961(n)) = A076345(n).
a(A005117(n)) = A076347(n).
a(A000290(n)) = A076349(n).

A076347 A076340(A005117(n)), real part of squarefree numbers mapped as defined in A076340, A076341.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 8, 12, 12, 16, 17, 16, 20, 31, 24, 24, 24, 28, 34, 32, 47, 32, 33, 36, 40, 49, 40, 62, 44, 48, 48, 65, 52, 49, 79, 56, 60, 60, 64, 47, 94, 68, 95, 66, 72, 72, 72, 95, 98, 80, 80, 84, 63, 88, 113, 88, 97, 127, 96, 81, 96, 100, 130, 104, 136, 104, 108, 108

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

			Applying the map as defined in A076340, A076341:
A005117(40)=65=5*13=(4+1)*(12+1) -> (4,1)*(12,1) = (4*12-1,12+4) = (47,16), therefore a(40)=47 and A076348(40)=16.

Imaginary part = A076348, A076342, A076343, A076349.

A076344 A076341(A001358(n)), imaginary part of semiprimes mapped as defined in A076340, A076341.

Original entry on oeis.org

0, -2, -8, 2, -2, 0, -12, -2, 8, 2, -16, 2, 4, -2, -8, -2, -16, -12, 8, -24, 2, -2, 16, -28, 2, -20, 2, 20, -2, -24, -4, -36, -2, 16, 2, -32, 20, -2, -8, -24, 2, -36, -48, -28, -2, -52, -2, 0, 32, 2, 28, -2, -48, -32, -2, 24, -64, 2, -56, 40, -4, 2, -72, 2, -20, 44, -2, -32, -76, -2

Offset: 1

Reinhard Zumkeller, Oct 08 2002

sign

			Applying the map as defined in A076340, A076341:
A001358(15)=39=3*13=(4-1)*(12+1) -> (4,-1)*(12,1) = (4*12+1,4-12) = (49,-8), therefore a(15)=-8 and A076343(15)=49.

Real part = A076343, A070750, A076348.

cp's OEIS Frontend

A076340 Real part of the function defined multiplicatively on the complex numbers by 2->(2,0) and p->((floor(p/4)+floor((p mod 4)/2))*4,2-(p mod 4)) for odd primes p.

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A076347 A076340(A005117(n)), real part of squarefree numbers mapped as defined in A076340, A076341.

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A076344 A076341(A001358(n)), imaginary part of semiprimes mapped as defined in A076340, A076341.

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