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%I #17 Sep 18 2020 11:24:20
%S 1,1,2,4,8,14,30,36,64,110,270,252,792,1008,1440,1344,5376,3936,14688,
%T 11664,19760,35200,96000,50880,97152,192192,145152,239904,917280,
%U 498240,2332800,864000,2334720,4300800,4257792,3172608
%N Number of k's such that A002034(k) = n.
%H Paul Erdős, S. W. Graham, Alexsandr Ivić, and Carl Pomerance, <a href="http://people.cst.cmich.edu/graha1sw/Pub/Papers/divfactorial.pdf">On the number of divisors of n!</a>, Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, ed. by B. C. Berndt, H. G. Diamond, A. J. Hildebrand, Birkhauser 1996, pp. 337-355.
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/SmarandacheFunction.html">MathWorld: Smarandache Function</a>
%F a(n) = A027423(n)-A027423(n-1) = A000005(A000142(n))-A000005(A000142(n-1)) i.e., number of divisors of n! which are not divisors of (n-1)! [for n>1]. - _Henry Bottomley_, Oct 22 2001
%F Erdős, Graham, Ivić, & Pomerance show that the average order of log a(n) is c log n/(log log n)^2 with c around 0.6289. - _Charles R Greathouse IV_, Jul 21 2015
%t a[n_] := DivisorSigma[0, n!] - DivisorSigma[0, (n-1)!]; a[1] = 1;
%t Array[a, 36] (* _Jean-François Alcover_, Sep 17 2020 *)
%o (PARI) a(n)=numdiv(n!)-numdiv((n-1)!) \\ _Charles R Greathouse IV_, Jul 21 2015
%Y Cf. A046021.
%K nonn
%O 1,3
%A _Christian G. Bower_
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