A047983 - OEIS

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

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OFFSET

1,5

COMMENTS

Invented by the HR concept formation program.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.

S. Colton, HR - Automatic Theory Formation in Pure Mathematics

FORMULA

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

EXAMPLE

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

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 *)

PROG

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

(Haskell)

a047983 n = length [x | x <- [1..n-1], a000005 x == a000005 n]

-- Reinhard Zumkeller, Nov 06 2011

(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

CROSSREFS

Position of the 0's form A007416. Cf. A005179.

Cf. A000005.