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

1,5

Comments

Invented by the HR concept formation program.

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]

Formula

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

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

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. A000005, A005179, A067004.

Sequence in context: A246370 A378125 A362557 * A070812 A308230 A061865

Adjacent sequences: A047980 A047981 A047982 * A047984 A047985 A047986

Keyword

nice,nonn,changed

Author

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

Status

approved