A060742 - OEIS

%I #20 Jun 15 2024 05:49:43

%S 0,0,1,2,4,9,15,27,41,68,111,218,328,624,929,1518,2016,3689,4965,9252,

%T 13177,20016,30697,56749,69434,94242,149558,190292,258370,492924,

%U 615063,1149403,1325124,1841343,2737190,3592273,4193855,8216492,12668800,17654339,20368544

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

%H 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 a(n) = A060741(n!/2) for n >= 2. - _Amiram Eldar_, Jun 15 2024

%e 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 f:= proc(n) local D,L;

%p   D:= numtheory:-divisors(n!);

%p   L:= sort(convert(D,list));

%p   nops(convert(L[2..-1]-L[1..-2],set) intersect D);

%p end proc:

%p map(f, [$0..34]); # _Robert Israel_, Jul 03 2017

%t a[n_ ] := Length[Intersection[Drop[d=Divisors[n! ], 1]-Drop[d, -1], d]]

%o (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 Cf. A000142, A027423, A060737, A060738, A060741.

%K nonn

%O 0,4

%A _Labos Elemer_, Apr 23 2001

%E Edited by _Dean Hickerson_, Jan 22 2002

%E One more term from _Robert G. Wilson v_, Jan 29 2002

%E a(33)-a(35) from _Robert Israel_, Jul 03 2017

%E a(36)-a(40) from _Amiram Eldar_, Jun 15 2024