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A336475
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Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (e+1)*p^e.
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5
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1, 1, 6, 1, 10, 6, 14, 1, 27, 10, 22, 6, 26, 14, 60, 1, 34, 27, 38, 10, 84, 22, 46, 6, 75, 26, 108, 14, 58, 60, 62, 1, 132, 34, 140, 27, 74, 38, 156, 10, 82, 84, 86, 22, 270, 46, 94, 6, 147, 75, 204, 26, 106, 108, 220, 14, 228, 58, 118, 60, 122, 62, 378, 1, 260, 132, 134, 34, 276, 140, 142, 27, 146, 74, 450, 38, 308, 156
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OFFSET
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1,3
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COMMENTS
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Although for all i, j: A003602(i) = A003602(j) => a(i) = a(j), it is not true that a(i) = a(j) => A003602(i) = A003602(j), because A038040 has a duplicate occurrence of a term on at least two odd positions: A038040(1875) = A038040(3125) = 18750.
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LINKS
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FORMULA
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Dirichlet g.f.: ((2^s - 2)^2/(4^s - 2^s)) * zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (n^2/12) * (2*log(n) + 4*gamma + 10*log(2)/3 - 1), where gamma is Euler's constant (A001620). (End)
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MATHEMATICA
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f[p_, e_] := (e+1)*p^e; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)
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PROG
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(PARI) A336475(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i, 1], 1, (1+f[i, 2]) * (f[i, 1]^f[i, 2]))); };
(Python)
from sympy import divisor_count
def A336475(n): return (m:=n>>(~n&n-1).bit_length())*divisor_count(m) # Chai Wah Wu, Jul 13 2022
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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