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Sum of the unitary divisors of n!.
1

%I #16 Jan 26 2023 21:35:08

%S 1,3,12,36,216,1020,8160,61920,507744,4383392,52600704,624249600,

%T 8739494400,109190390400,1583122968000,25318378008000,455730804144000,

%U 8193040840252800,163860816805056000,3256371347261760000,67204676251838361600,1366492477414792734720

%N Sum of the unitary divisors of n!.

%H Amiram Eldar, <a href="/A064028/b064028.txt">Table of n, a(n) for n = 1..450</a>

%H Charles R. Wall, <a href="https://www.fq.math.ca/Scanned/22-3/advanced22-3.pdf">Problem H-374</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 22, No. 3 (1984), p. 280; <a href="https://www.fq.math.ca/Scanned/24-2/advanced24-2.pdf">Bounds of Joy</a>, Solution to Problem H-374 by the proposer, ibid., Vol. 24, No. 2 (1986), p. 188.

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

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

%e 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.

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

%o (PARI) valp(n,p)=my(s); while(n\=p, s+=n); s

%o a(n)=my(s=1); forprime(p=2,n, s*=p^valp(n,p)+1); s \\ _Charles R Greathouse IV_, Jan 26 2023

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

%K nonn

%O 1,2

%A _Labos Elemer_, Sep 11 2001