A064028 Sum of the unitary divisors of n!.

1, 3, 12, 36, 216, 1020, 8160, 61920, 507744, 4383392, 52600704, 624249600, 8739494400, 109190390400, 1583122968000, 25318378008000, 455730804144000, 8193040840252800, 163860816805056000, 3256371347261760000, 67204676251838361600, 1366492477414792734720

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..450

Charles R. Wall, Problem H-374, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 22, No. 3 (1984), p. 280; Bounds of Joy, Solution to Problem H-374 by the proposer, ibid., Vol. 24, No. 2 (1986), p. 188.

Formula

a(n) = usigma(n!) = A034448(A000142(n)).

a(n)/n! <= 2 (while usigma(n)/n and sigma(n!)/n! are unbounded; Wall, 1984). - Amiram Eldar, Feb 08 2022

Example

n=6, 6! = 720, sum of the 8 unitary ones of its 30 divisors is 1020, a(6) = 720+1+16+45+9+80+5+144 = 1020.

Mathematica

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma/@ (Range[17]!) (* Amiram Eldar, Jun 23 2019 *)

Prog

(PARI) valp(n, p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2, n, s*=p^valp(n, p)+1); s \\ Charles R Greathouse IV, Jan 26 2023

Crossrefs

Cf. A034448, A003444, A046656, A056657, A056171, A056172, A000203, A000142, A062569.

Sequence in context: A282022 A004661 A074430 * A326241 A110950 A102744

Adjacent sequences: A064025 A064026 A064027 * A064029 A064030 A064031

Keyword

nonn

Author

Labos Elemer, Sep 11 2001

Status

approved