cp's OEIS Frontend

A331592 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.

%I A331592 #16 Jan 25 2020 08:02:03
%S A331592 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 A331592 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 A331592 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 A331592 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 A331592 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 A331592 The result depends only on the prime signature of n.
%C A331592 k first appears at A191555(k).
%H A331592 Antti Karttunen, <a href="/A331592/b331592.txt">Table of n, a(n) for n = 1..65537</a>
%H A331592 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A331592 a(n) = min(A001221(n), A331591(n)) = min(A001221(n), A001221(A293442(n))).
%F A331592 a(A225546(n)) = a(n).
%F A331592 a(A003961(n)) = a(n).
%F A331592 a(n^2) = a(n).
%e A331592 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 A331592 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 A331592 (PARI) A331592(n) = min(omega(n), A331591(n)); \\ Uses also code from A331591.
%Y A331592 Cf. A000961, A001221, A191555, A293442, A329332.
%Y A331592 Sequences with related definitions: A331308, A331591, A331593.
%Y A331592 A003961, A225546 are used to express relationship between terms of this sequence.
%Y A331592 Differs from = A071625 for the first time at n=216, where a(216) = 2, while A071625(216) = 1.
%K A331592 nonn
%O A331592 1,12
%A A331592 _Antti Karttunen_ and _Peter Munn_, Jan 21 2020