A369759 The sum of unitary divisors of the smallest cubefull exponentially odd number that is divisible by n.

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 33, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 33, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A034448(A356192(n)).

Multiplicative with a(p) = p^3 + 1, a(p^e) = p^e + 1 for an odd e >= 3, and a(p^e) = p^(e+1) + 1 for an even e.

a(n) >= A034448(n), with equality if and only if n is cubefull exponentially odd number (A335988).

Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(2*s-2) - 1/p^(3*s-5) + 1/p^(4*s-5) - 1/p^(4*s-3)).

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = (zeta(4)*zeta(6)/zeta(2)) * Product_{p prime} (1 - 1/p^6 + 1/p^11 - 1/p^12) = 0.65813930591740259189... .

Mathematica

f[p_, e_] := p^If[OddQ[e], Max[e, 3], e+1] + 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~, 1 + f[i, 1]^if(f[i, 2]%2, max(f[i, 2], 3), f[i, 2] + 1)); }

Crossrefs

Cf. A034448, A335988, A356192, A365480, A369757, A369758.

Cf. A013661, A013662, A013664.

Sequence in context: A328640 A351266 A369721 * A033479 A254139 A034116

Adjacent sequences: A369756 A369757 A369758 * A369760 A369761 A369762

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 31 2024

Status

approved