A366765 - OEIS

%I #8 Oct 21 2023 11:37:54

%S 1,2,3,2,5,6,7,8,3,10,11,6,13,14,15,16,17,6,19,10,21,22,23,24,5,26,27,

%T 14,29,30,31,32,33,34,35,6,37,38,39,40,41,42,43,22,15,46,47,48,7,10,

%U 51,26,53,54,55,56,57,58,59,30,61,62,21,64,65,66,67,34,69

%N The largest divisor of n that have no exponent 2 in their prime factorization.

%C The largest term of A337050 that divides n.

%C The number of these divisors is A366763(n), and their sum is A366764(n).

%H Amiram Eldar, <a href="/A366765/b366765.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p if e <= 2 and p^e otherwise.

%F a(n) <= n, with equality if and only if n is in A337050.

%F Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2)).

%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 2/p^3 - 1/p^4) = 0.83234421330425224469... .

%t f[p_, e_] := p^If[e < 3, 1, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1] ^ if(f[i, 2] < 3, 1, f[i, 2]));}

%Y Cf. A337050, A366763, A366764.

%Y Similar sequences: A055231, A057521, A008833, A350390.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Oct 21 2023