A392090 - OEIS

%I #8 Dec 30 2025 11:41:08

%S 1,7,26,7,124,182,342,7,26,868,1330,182,2196,2394,3224,15,4912,182,

%T 6858,868,8892,9310,12166,182,124,15372,26,2394,24388,22568,29790,31,

%U 34580,34384,42408,182,50652,48006,57096,868,68920,62244,79506,9310,3224,85162,103822

%N The unitary totient of the smallest cubefull number that is divisible by n.

%C First differs from the sequence of absolute values of A063453 at n = 16.

%H Amiram Eldar, <a href="/A392090/b392090.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A047994(A356193(n)).

%F Multiplicative with a(p^e) = p^max(e, 3) - 1.

%F Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(s-1) - 2/p^s - 1/p^(2*s-4) + 2/p^(2*s-1) + 1/p^(4*s-4) - 1/p^(4*s-3)).

%F Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 - 2/p^4 + 3/p^5 + 2/p^7 - 2/p^8 + 1/p^12 - 2/p^13 + 1/p^14) = 0.58988140992095483236... .

%t f[p_, e_] := p^Max[e, 3] - 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~, f[i, 1]^max(f[i, 2], 3)-1);}

%Y Cf. A002117, A013662, A036966, A047994, A063453, A356193, A390756.

%K nonn,mult,easy

%O 1,2

%A _Amiram Eldar_, Dec 30 2025