A356191 - OEIS

%I #14 Sep 09 2023 07:42:39

%S 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,

%T 26,27,56,29,30,31,32,33,34,35,216,37,38,39,40,41,42,43,88,135,46,47,

%U 96,343,250,51,104,53,54,55,56,57,58,59,120,61,62,189,128,65

%N a(n) is the smallest exponentially odd number that is divisible by n.

%H Amiram Eldar, <a href="/A356191/b356191.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.

%F a(n) = n iff n is in A268335.

%F a(n) = A064549(n)/A007913(n).

%F a(n) = n*A336643(n).

%F a(n) = n^2/A350390(n).

%F From _Vaclav Kotesovec_, Sep 09 2023: (Start)

%F 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)).

%F 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 f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,

%F f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...

%F and gamma is the Euler-Mascheroni constant A001620. (End)

%t f[p_, e_] := If[OddQ[e], p^e, p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (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)))};

%o (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

%Y Cf. A064549, A007913, A268335, A336643, A350390.

%Y Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356192, A356193, A356194.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Jul 29 2022