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

%I A047983 #52 Feb 04 2025 07:13:47
%S A047983 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,
%T A047983 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,
%U A047983 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
%N A047983 Number of integers less than n but with the same number of divisors.
%C A047983 Invented by the HR concept formation program.
%H A047983 T. D. Noe, <a href="/A047983/b047983.txt">Table of n, a(n) for n = 1..10000</a>
%H A047983 Simon Colton, <a href="http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
%H A047983 Simon Colton, <a href="http://web.archive.org/web/20070831060523/http://www.dai.ed.ac.uk/homes/simonco/research/hr/">HR - Automatic Theory Formation in Pure Mathematics</a>, 1998-1999. [Wayback Machine link]
%F A047983 f(n) = |{k < n : tau(k) = tau(n)}|.
%F A047983 a(n) = A067004(n) - 1. - _Amiram Eldar_, Feb 04 2025
%e A047983 f(10) = 2 because tau(10) = 4 and also tau(6) = tau(8) = 4.
%t A047983 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 *)
%t A047983 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 *)
%o A047983 (PARI) A047983(n) = {local(d);d=numdiv(n);sum(k=1,n-1,(numdiv(k)==d))} \\ _Michael B. Porter_, Mar 01 2010
%o A047983 (Haskell)
%o A047983 a047983 n = length [x | x <- [1..n-1], a000005 x == a000005 n]
%o A047983 -- _Reinhard Zumkeller_, Nov 06 2011
%o A047983 (Python)
%o A047983 from sympy import divisor_count as D
%o A047983 def a(n): return sum([1 for k in range(1, n) if D(k) == D(n)]) # _Indranil Ghosh_, Apr 30 2017
%Y A047983 Position of the 0's form A007416.
%Y A047983 Cf. A000005, A005179, A067004.
%K A047983 nice,nonn
%O A047983 1,5
%A A047983 Simon Colton (simonco(AT)cs.york.ac.uk)