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A060742 Number of divisors of n! which are also differences between consecutive divisors of n! (ordered by size).

%I A060742 #20 Jun 15 2024 05:49:43
%S A060742 0,0,1,2,4,9,15,27,41,68,111,218,328,624,929,1518,2016,3689,4965,9252,
%T A060742 13177,20016,30697,56749,69434,94242,149558,190292,258370,492924,
%U A060742 615063,1149403,1325124,1841343,2737190,3592273,4193855,8216492,12668800,17654339,20368544
%N A060742 Number of divisors of n! which are also differences between consecutive divisors of n! (ordered by size).
%H A060742 Daniel Berend and J. E. Harmse, <a href="http://dx.doi.org/10.5802/aif.1348">Gaps between consecutive divisors of factorials</a>, Ann. Inst. Fourier, 43 (3) (1993), 569-583.
%F A060742 a(n) = A060741(n!/2) for n >= 2. - _Amiram Eldar_, Jun 15 2024
%e A060742 For n = 5, n! = 120; divisors = {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}; differences = {1,1,1,1,1,2,2,2,3,5,4,6,10,20,60}; intersection = {1,2,3,4,5,6,10,20,60}, so a(5) = 9.
%p A060742 f:= proc(n) local D,L;
%p A060742   D:= numtheory:-divisors(n!);
%p A060742   L:= sort(convert(D,list));
%p A060742   nops(convert(L[2..-1]-L[1..-2],set) intersect D);
%p A060742 end proc:
%p A060742 map(f, [$0..34]); # _Robert Israel_, Jul 03 2017
%t A060742 a[n_ ] := Length[Intersection[Drop[d=Divisors[n! ], 1]-Drop[d, -1], d]]
%o A060742 (PARI) a(n) = {my(v = List(), f = n!, d1 = 1, del); fordiv(f, d, if(d > 1, del = d - d1; if(!(f % del), listput(v, del)); d1 = d)); #Set(v);} \\ _Amiram Eldar_, Jun 15 2024
%Y A060742 Cf. A000142, A027423, A060737, A060738, A060741.
%K A060742 nonn
%O A060742 0,4
%A A060742 _Labos Elemer_, Apr 23 2001
%E A060742 Edited by _Dean Hickerson_, Jan 22 2002
%E A060742 One more term from _Robert G. Wilson v_, Jan 29 2002
%E A060742 a(33)-a(35) from _Robert Israel_, Jul 03 2017
%E A060742 a(36)-a(40) from _Amiram Eldar_, Jun 15 2024