cp's OEIS Frontend

A340682 The closure under squaring of the nonunit squarefree numbers.

%I A340682 #18 Jun 01 2025 16:37:40
%S A340682 2,3,4,5,6,7,9,10,11,13,14,15,16,17,19,21,22,23,25,26,29,30,31,33,34,
%T A340682 35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,65,66,67,69,
%U A340682 70,71,73,74,77,78,79,81,82,83,85,86,87,89,91,93,94,95,97,100,101,102,103,105,106,107,109,110,111,113
%N A340682 The closure under squaring of the nonunit squarefree numbers.
%C A340682 Numbers of the form s^(2^e), where s is a nonunit squarefree number, and e >= 0.
%C A340682 The categorization provided by this sequence and its complement, A340681, is an alternative extension (to all integers greater than 1) of the 2-way distinction between squarefree and nonsquarefree as it applies to nonsquares.
%C A340682 All positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. This sequence lists the numbers where this factorization has only one term, that is numbers m such that A331591(m) = 1.
%C A340682 Presence in the sequence is determined by prime signature. The set of represented signatures starts: {{1}, {2}, {1,1}, {1,1,1}, {4}, {2,2}, {1,1,1,1}, {1,1,1,1,1}, {2,2,2}, {1,1,1,1,1,1}, {1,1,1,1,1,1,1}, {8}, {4,4}, {2,2,2,2}, {1,1,1,1,1,1,1,1}, ...}. Representing each signature in the set by the least number with that signature, we get the set A133492.
%C A340682 Positions of terms > 1 in A340675.
%H A340682 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%e A340682 12 = 3 * 4 = 3^1 * 2^2 = 3^(2^0) * 2^(2^1). This is the (unique) factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. As this factorization has 2 terms, 12 is not in the sequence.
%e A340682 The equivalent factorization for 36 is 36 = 6^2 = 6^(2^1). As this factorization has only 1 term, 36 is in the sequence.
%t A340682 Select[Range[2, 120], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] == 2^IntegerExponent[u[[1]], 2] &] (* _Amiram Eldar_, Feb 13 2021 *)
%o A340682 (PARI) isA340682(n) = if(!issquare(n), issquarefree(n), (n>1)&&isA340682(sqrtint(n)));
%o A340682 (Python)
%o A340682 from math import isqrt
%o A340682 from sympy import mobius, integer_nthroot
%o A340682 def A340682(n):
%o A340682     def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A340682     def f(x): return int(n+x-sum(g(integer_nthroot(x,1<<e)[0])-1 for e in range(x.bit_length().bit_length())))
%o A340682     m, k = n, f(n)
%o A340682     while m != k: m, k = k, f(k)
%o A340682     return m # _Chai Wah Wu_, Jun 01 2025
%Y A340682 Cf. A340675.
%Y A340682 Cf. A340681 (complement, apart from 1 which is in neither).
%Y A340682 Subsequence of A072774, A210490.
%Y A340682 Positions of ones in A331591.
%Y A340682 Union of A005117 \ {1} and A340674.
%Y A340682 Cf. subsequences: A050376, A133492.
%K A340682 nonn,easy
%O A340682 1,1
%A A340682 _Antti Karttunen_ and _Peter Munn_, Feb 07 2021