M. Sinan Kul - cp's OEIS frontend

User: M. Sinan Kul

M. Sinan Kul has authored 2 sequences.

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

M. Sinan Kul, Apr 11 2017

nonn

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

Project Euler, Problem 598: Split Divisibilities

Cf. A000005.

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 Leonid Bedratyuk, Apr 16 2017

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

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

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

A272268 Records in A001481 that are more than twice the previous record.

Original entry on oeis.org

1, 4, 9, 20, 41, 85, 173, 349, 701, 1405, 2813, 5629, 11261, 22525, 45053, 90113, 180232, 360466, 720937, 1441877, 2883761, 5767525, 11535053, 23070112, 46140228, 92280457, 184560921, 369121849, 738243700, 1476487402, 2952974809, 5905949620, 11811899241

Offset: 1

M. Sinan Kul, Apr 24 2016

nonn

This list can be seen as the r^2 (square of the radius r) of the homocentric circles that are centered at the origin and pass through at least 4 lattice points, the innermost circle being the unit circle.
If we start with the unit circle (x^2 + y^2 = 1), the smallest circle that pass through at least four lattice points would be x^2 + y^2 = 4 with (2,0), (0,2), (-2,0), (0,-2). Similarly next circle would be x^2 + y^2 = 9 passing through (3,0), (0,3), (-3,0), (0,-3), and the next x^2 + y^2 = 20 passing through (2,4), (4,2), (-2,4), (-4,2), (-2,-4), (-4,-2), (2,-4), (4,-2), etc.
It is also worth mentioning that a square can be drawn with vertexes on the lattice points of a circle and the sides of that square wouldn't touch the smaller circle.

M. Sinan Kul, PARI/GP code

Cf. A001481.

NestList[SelectFirst[Range[2 # + 1, 5 #], SquaresR[2, #] > 0 &] &, 1, 25] (* Michael De Vlieger, Apr 25 2016, Version 10 *)

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User: M. Sinan Kul

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

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