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a(n) is the smaller of the number of terms in the factorizations of n into (1) powers of distinct primes and (2) powers of squarefree numbers with distinct exponents that are powers of 2.
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%I #16 Jan 25 2020 08:02:03

%S 0,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,1,1,1,1,1,

%T 1,1,1,1,1,2,1,1,1,2,2,1,1,2,1,2,1,2,1,2,1,2,1,1,1,2,1,1,2,1,1,1,1,2,

%U 1,1,1,2,1,1,2,2,1,1,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,2,2,1,1,1,1,2,1

%N a(n) is the smaller of the number of terms in the factorizations of n into (1) powers of distinct primes and (2) powers of squarefree numbers with distinct exponents that are powers of 2.

%C See A329332 for a description of the relationship between the two factorizations. From this relationship we get the formula a(n) = min(A001221(n), A001221(A225546(n))).

%C The result depends only on the prime signature of n.

%C k first appears at A191555(k).

%H Antti Karttunen, <a href="/A331592/b331592.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = min(A001221(n), A331591(n)) = min(A001221(n), A001221(A293442(n))).

%F a(A225546(n)) = a(n).

%F a(A003961(n)) = a(n).

%F a(n^2) = a(n).

%e The factorization of 6 into powers of distinct primes is 6 = 2^1 * 3^1 = 2 * 3, which has 2 terms. Its factorization into powers of squarefree numbers with distinct exponents that are powers of 2 is 6 = 6^(2^0) = 6^1, which has 1 term. So a(6) is min(2,1) = 1.

%e The factorization of 40 into powers of distinct primes is 40 = 2^3 * 5^1 = 8 * 5, which has 2 terms. Its factorization into powers of squarefree numbers with distinct exponents that are powers of 2 is 40 = 10^(2^0) * 2^(2^1) = 10^1 * 2^2 = 10 * 4, which has 2 terms. So a(40) is min(2,2) = 2.

%o (PARI) A331592(n) = min(omega(n), A331591(n)); \\ Uses also code from A331591.

%Y Cf. A000961, A001221, A191555, A293442, A329332.

%Y Sequences with related definitions: A331308, A331591, A331593.

%Y A003961, A225546 are used to express relationship between terms of this sequence.

%Y Differs from = A071625 for the first time at n=216, where a(216) = 2, while A071625(216) = 1.

%K nonn

%O 1,12

%A _Antti Karttunen_ and _Peter Munn_, Jan 21 2020