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A339514 Number of subsets of {1..n} whose elements have the same number of divisors.

%I A339514 #15 Feb 16 2025 08:34:01
%S A339514 1,2,3,5,6,10,11,19,21,23,27,43,44,76,84,100,101,165,167,295,299,331,
%T A339514 395,651,652,656,784,1040,1048,1560,1562,2586,2602,3114,4138,6186,
%U A339514 6187,8235,12331,20523,20527,24623,24631,32823,32855,32919,49303,65687,65688
%N A339514 Number of subsets of {1..n} whose elements have the same number of divisors.
%H A339514 Sebastian Karlsson, <a href="/A339514/b339514.txt">Table of n, a(n) for n = 0..1000</a>
%H A339514 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Divisor.html">Divisor</a>
%F A339514 a(0) = 1, a(n) = a(n-1) + 2^A047983(n). - _Sebastian Karlsson_, Dec 26 2020
%e A339514 a(8) = 21 subsets: {}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {2, 3}, {2, 5}, {2, 7}, {3, 5}, {3, 7}, {5, 7}, {6, 8}, {2, 3, 5}, {2, 3, 7}, {2, 5, 7}, {3, 5, 7} and {2, 3, 5, 7}.
%o A339514 (Python)
%o A339514 from sympy import divisors
%o A339514 def test(n):
%o A339514     if n<2: return n-1
%o A339514     return len(divisors(n))
%o A339514 def a(n):
%o A339514     tests = [test(i) for i in range(n+1)]
%o A339514     return sum(2**tests.count(v)-1 for v in set(tests))
%o A339514 print([a(n) for n in range(49)]) # _Michael S. Branicky_, Dec 07 2020
%Y A339514 Cf. A000005, A339511, A339512, A047983.
%K A339514 nonn
%O A339514 0,2
%A A339514 _Ilya Gutkovskiy_, Dec 07 2020
%E A339514 a(25)-a(48) from _Michael S. Branicky_, Dec 07 2020