A277621 Number of pairs (a,b) such that a*b = n! and d(a) = d(b) with d = A000005 and a <= b.

1, 1, 0, 1, 0, 1, 1, 1, 0, 5, 3, 5, 5, 13, 11, 11, 11, 13, 45, 105, 136, 105, 165, 332, 492, 501, 482, 684, 720, 1095, 1656, 3273, 3136, 3901, 4948, 6674, 7641, 15047, 12879, 17217, 38901, 75540, 37743, 73594, 84249, 88592, 207324, 403493, 710536, 922853, 662019

Offset

0,10

Links

Table of n, a(n) for n=0..50.

Project Euler, Problem 598: Split Divisibilities

Example

For n = 9, there are 5 pairs (a,b): (384,945), (420,864), (480,756), (540,672), (560,648)

Maple

a:=proc(n) local S, nf, DD, d, dd:with(numtheory): S:=0:nf:=n!:DD:=divisors(nf):dd:=floor(sqrt(nf)): for d in DD while d <dd do  if tau(d)=tau(nf/d) then S:=S +1 end if: end do:  S end proc: # only for small n # Leonid Bedratyuk, Apr 16 2017

Mathematica

a[n_] := Length@ Select[ Divisors[n!], # <= n!/# && Equal @@ DivisorSigma[0, {#, n!/#}] &]; a /@ Range[0, 20] (* Giovanni Resta, Apr 11 2017 *)

Prog

(PARI) a(n)=my(c=0); fordiv(n!, a, my(b=n!/a); if(a>b, break); if( numdiv(a) == numdiv(b), c++)); c

Crossrefs

Cf. A000005.

Sequence in context: A194584 A073316 A245979 * A228046 A092516 A269893

Adjacent sequences: A277618 A277619 A277620 * A277622 A277623 A277624

Keyword

nonn

Author

M. Sinan Kul, Apr 11 2017

Extensions

a(26)-a(50) from Giovanni Resta, Apr 11 2017

Status

approved