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Number of subsets of {1..n} whose elements have the same number of distinct prime factors.
3

%I #22 Jul 29 2023 06:44:16

%S 1,2,3,5,9,17,18,34,66,130,132,260,264,520,528,544,1056,2080,2112,

%T 4160,4224,4352,4608,8704,9216,17408,18432,34816,36864,69632,69633,

%U 135169,266241,270337,278529,294913,327681,589825,655361,786433,1048577,1572865,1572867

%N Number of subsets of {1..n} whose elements have the same number of distinct prime factors.

%H Sebastian Karlsson, <a href="/A339512/b339512.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>

%F a(n) = 1 + Sum_{k=1..n} 2^A334655(k). - _Sebastian Karlsson_, Feb 18 2021

%e a(5) = 17 subsets: {}, {1}, {2}, {3}, {4}, {5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5} and {2, 3, 4, 5}.

%o (Python)

%o from sympy import primefactors

%o def test(n):

%o if n==0: return -1

%o return len(primefactors(n))

%o def a(n):

%o tests = [test(i) for i in range(n+1)]

%o return sum(2**tests.count(v)-1 for v in set(tests))

%o print([a(n) for n in range(43)]) # _Michael S. Branicky_, Dec 07 2020

%Y Cf. A001221, A339511, A339514.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Dec 07 2020

%E a(23)-a(42) from _Michael S. Branicky_, Dec 07 2020