A331592 - OEIS

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.

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, 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, 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

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OFFSET

1,12

COMMENTS

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))).

The result depends only on the prime signature of n.

k first appears at A191555(k).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

FORMULA

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

a(A225546(n)) = a(n).

a(A003961(n)) = a(n).

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

EXAMPLE

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.

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.

PROG

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

CROSSREFS

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

Sequences with related definitions: A331308, A331591, A331593.

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

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

Sequence in context: A088323 A003652 A071625 * A353745 A309004 A355382

Adjacent sequences: A331589 A331590 A331591 * A331593 A331594 A331595

KEYWORD

nonn

AUTHOR

Antti Karttunen and Peter Munn, Jan 21 2020

STATUS

approved