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A351465
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Let f be multiplicative with f(prime(k)^e) = k + e*i for any k, e > 0 (where i denotes the imaginary unit); a(n) is the imaginary part of f(n). See A351464 for the real part.
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2
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0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 5, 4, 1, 4, 1, 7, 6, 6, 1, 7, 2, 7, 3, 9, 1, 10, 1, 5, 7, 8, 7, 6, 1, 9, 8, 10, 1, 13, 1, 11, 8, 10, 1, 9, 2, 5, 9, 13, 1, 5, 8, 13, 10, 11, 1, 15, 1, 12, 10, 6, 9, 16, 1, 15, 11, 18, 1, 8, 1, 13, 7, 17, 9, 19, 1, 13
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OFFSET
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1,4
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LINKS
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EXAMPLE
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For n = 42:
- 42 = 2 * 3 * 7 = prime(1)^1 * prime(2)^1 * prime(4)^1,
- f(42) = (1+i) * (2+i) * (4+i) = 1 + 13*i,
- and a(42) = 13.
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MAPLE
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b:= proc(n) option remember; uses numtheory;
mul(pi(i[1])+i[2]*I, i=ifactors(n)[2])
end:
a:= n-> Im(b(n)):
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MATHEMATICA
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f[p_, e_] := PrimePi[p] + e*I; a[1] = 0; a[n_] := Im[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Feb 15 2022 *)
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PROG
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(PARI) a(n) = { my (f=factor(n), p=f[, 1]~, e=f[, 2]~); imag(prod (k=1, #p, primepi(p[k]) + I*e[k])) }
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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