A047983 Number of integers less than n but with the same number of divisors.

0, 0, 1, 0, 2, 0, 3, 1, 1, 2, 4, 0, 5, 3, 4, 0, 6, 1, 7, 2, 5, 6, 8, 0, 2, 7, 8, 3, 9, 1, 10, 4, 9, 10, 11, 0, 11, 12, 13, 2, 12, 3, 13, 5, 6, 14, 14, 0, 3, 7, 15, 8, 15, 4, 16, 5, 17, 18, 16, 0, 17, 19, 9, 0, 20, 6, 18, 10, 21, 7, 19, 1, 20, 22, 11, 12, 23, 8, 21, 1, 1, 24, 22, 2, 25, 26, 27

Offset: 1

Author

Simon Colton (simonco(AT)cs.york.ac.uk)

Keywords

• nice
• nonn

Comments

Invented by the HR concept formation program.

Examples

			f(10) = 2 because tau(10) = 4 and also tau(6) = tau(8) = 4.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
• Simon Colton, HR - Automatic Theory Formation in Pure Mathematics, 1998-1999. [Wayback Machine link]

Crossrefs

Position of the 0's form A007416.

Cf. A000005, A005179, A067004.

Programs

Haskell

a047983 n = length [x | x <- [1..n-1], a000005 x == a000005 n]
-- Reinhard Zumkeller, Nov 06 2011

Mathematica

a[n_] := With[{tau = DivisorSigma[0, n]}, Length[ Select[ Range[n-1], DivisorSigma[0, #] == tau & ]]]; Table[a[n], {n, 1, 87}] (* Jean-François Alcover, Nov 30 2011 *)
Module[{nn=90,ds},ds=DivisorSigma[0,Range[nn]];Table[Count[Take[ds,n], ds[[n]]]- 1,{n,nn}]] (* Harvey P. Dale, Feb 16 2014 *)

PARI

A047983(n) = {local(d);d=numdiv(n);sum(k=1,n-1,(numdiv(k)==d))} \\ Michael B. Porter, Mar 01 2010

Python

from sympy import divisor_count as D
def a(n): return sum([1 for k in range(1, n) if D(k) == D(n)]) # Indranil Ghosh, Apr 30 2017

Formula

f(n) = |{k < n : tau(k) = tau(n)}|.

a(n) = A067004(n) - 1. - Amiram Eldar, Feb 04 2025