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A356191 a(n) is the smallest exponentially odd number that is divisible by n. 12
1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 32, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 96, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A268335.
a(n) = A064549(n)/A007913(n).
a(n) = n*A336643(n).
a(n) = n^2/A350390(n).
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(6-5*s) + p^(7-5*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(3-3*s) - p^(4-3*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(2) * n^2 / 24 * (log(n) + 3*gamma - 1/2 + 12*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...
and gamma is the Euler-Mascheroni constant A001620. (End)
MATHEMATICA
f[p_, e_] := If[OddQ[e], p^e, p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]%2, f[i, 1]^f[i, 2], f[i, 1]^(f[i, 2]+1)))};
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023
CROSSREFS
Sequence in context: A268675 A268385 A093928 * A135874 A372329 A138682
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jul 29 2022
STATUS
approved

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Last modified May 26 14:50 EDT 2024. Contains 372826 sequences. (Running on oeis4.)