A076344 -id:A076344 - cp's OEIS frontend

A076341 Imaginary 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.

0, 0, -1, 0, 1, -2, -1, 0, -8, 2, -1, -4, 1, -2, 0, 0, 1, -16, -1, 4, -12, -2, -1, -8, 8, 2, -47, -4, 1, 0, -1, 0, -16, 2, 4, -32, 1, -2, -8, 8, 1, -24, -1, -4, -17, -2, -1, -16, -16, 16, -12, 4, 1, -94, 8, -8, -24, 2, -1, 0, 1, -2, -79, 0, 16, -32, -1, 4, -28, 8, -1, -64, 1, 2, 17, -4

Offset: 1

Reinhard Zumkeller, Oct 08 2002

sign

			n=21: 21 = 3*7 = (4-1)*(8-1) = (4,-1)*(8,-1) -> (32-(-1)*(-1),-4+(-8)) = (31,-12), therefore a(21)=-12, A076340(21)=31;
n=35: 35 = 5*7 = (4+1)*(8-1) = (4,1)*(8,-1) -> (32-1*(-1),-4+8) = (33,4), therefore a(35)=4, A076340(35)=33.

Real part = A076340.
Cf. A070750, A076344, A076346, A076348, A076350, A076351.
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_] := Im[b[n]];
Array[a, 100] (* Jean-François Alcover, Dec 12 2021 *)

a(A000040(n)) = A070750(n).
a(A001358(n)) = A076344(n).
a(A000961(n)) = A076346(n).
a(A005117(n)) = A076348(n).
a(A000290(n)) = A076350(n);
a(A076351(n)) = 0.

A076348 A076341(A005117(n)), imaginary part of squarefree numbers mapped as defined in A076340, A076341.

Original entry on oeis.org

0, 0, -1, 1, -2, -1, 2, -1, 1, -2, 0, 1, -1, -12, -2, -1, 2, 1, 0, -1, -16, 2, 4, 1, -2, -8, 1, -24, -1, -2, -1, -12, 1, 8, -24, 2, -1, 1, -2, 16, -32, -1, -28, 8, -1, 1, 2, -20, -16, -1, 2, -1, 20, -2, -24, 1, -4, -36, -2, 16, 1, 1, -24, -1, -17, 2, -1, 1, 16, -32, 1, -48, 20, -2, -8

Offset: 1

Reinhard Zumkeller, Oct 08 2002

sign

			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)=16 and A076347(40)=47.

Real part = A076347, A070750, A076344, A076350, A076352.

A076343 A076340(A001358(n)), real part of semiprimes mapped as defined in A076340, A076341.

Original entry on oeis.org

4, 8, 15, 8, 16, 17, 31, 24, 15, 24, 47, 32, 33, 40, 49, 48, 63, 65, 49, 79, 56, 64, 47, 95, 72, 95, 80, 63, 88, 113, 97, 127, 96, 81, 104, 145, 97, 120, 129, 143, 120, 161, 175, 159, 136, 191, 144, 145, 111, 144, 129, 160, 209, 191, 168, 143, 239, 176, 241, 143

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

			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)=49 and A076344(120)=-8.

Imaginary part = A076344, A076342, A076347.

cp's OEIS Frontend

A076341 Imaginary 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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A076348 A076341(A005117(n)), imaginary part of squarefree numbers mapped as defined in A076340, A076341.

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A076343 A076340(A001358(n)), real part of semiprimes mapped as defined in A076340, A076341.

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