A369718 - OEIS

%I #12 Jan 30 2024 08:23:17

%S 1,5,10,5,26,50,50,9,10,130,122,50,170,250,260,17,290,50,362,130,500,

%T 610,530,90,26,850,28,250,842,1300,962,33,1220,1450,1300,50,1370,1810,

%U 1700,234,1682,2500,1850,610,260,2650,2210,170,50,130,2900,850,2810,140

%N The sum of unitary divisors of the smallest powerful number that is a multiple of n.

%H Amiram Eldar, <a href="/A369718/b369718.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A034448(A197863(n)).

%F Multiplicative with a(p) = p^2 + 1 and a(p^e) = p^e + 1 for e >= 2.

%F a(n) >= A034448(n), with equality if and only if n is powerful (A001694).

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

%F Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^6 - 2/p^7 + 1/p^8) = 0.73644353930922037459... .

%t f[p_, e_] := If[e == 1, p^2 + 1, 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] == 1, 1 + f[i,1]^2, 1 + f[i,1]^f[i,2]));}

%Y Cf. A034448, A001694, A197863, A369716, A369717, A369721.

%Y Cf. A002117, A013661, A183699.

%K nonn,easy,mult,look

%O 1,2

%A _Amiram Eldar_, Jan 30 2024