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A360902 Numbers with the same number of squarefree divisors and powerful divisors.

%I A360902 #12 Feb 26 2023 03:26:36
%S A360902 1,4,9,25,36,48,49,80,100,112,121,162,169,176,196,208,225,272,289,304,
%T A360902 361,368,405,441,464,484,496,529,567,592,656,676,688,720,752,841,848,
%U A360902 891,900,944,961,976,1008,1053,1072,1089,1136,1156,1168,1200,1225,1250,1264
%N A360902 Numbers with the same number of squarefree divisors and powerful divisors.
%C A360902 Numbers k such that A034444(k) = A005361(k).
%C A360902 Numbers whose squarefree kernel (A007947) and powerful part (A057521) have the same number of divisors (A000005).
%C A360902 If k and m are coprime terms, then k*m is also a term.
%C A360902 All the terms are exponentially 2^n-numbers (A138302).
%C A360902 The characteristic function of this sequence depends only on prime signature.
%C A360902 Numbers whose canonical prime factorization has exponents whose geometric mean is 2.
%C A360902 Equivalently, numbers of the form Product_{i=1..m} p_i^(2^k_i), where p_i are distinct primes, and Sum_{i=1..m} k_i = m (i.e., the exponents k_i have an arithmetic mean 1).
%C A360902 1 is the only squarefree (A005117) term.
%C A360902 Includes the squares of squarefree numbers (A062503), which are the powerful (A001694) terms of this sequence.
%C A360902 The squares of primes (A001248) are the only terms that are prime powers (A246655).
%C A360902 Numbers of the for m*p^(2^k), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k - 1, are all terms. In particular, this sequence includes numbers of the form p^4*q, where p != q are primes (A178739), and numbers of the form p^8*q*r where p, q, and r are distinct primes (A179747).
%C A360902 The corresponding numbers of squarefree (or powerful) divisors are 1, 2, 2, 2, 4, 4, 2, 4, 4, 4, 2, 4, ... . The least term with 2^k squarefree divisors is A360903(k).
%H A360902 Amiram Eldar, <a href="/A360902/b360902.txt">Table of n, a(n) for n = 1..10000</a>
%e A360902 4 is a term since it has 2 squarefree divisors (1 and 2) and 2 powerful divisors (1 and 4).
%e A360902 36 is a term since it has 4 squarefree divisors (1, 2, 3 and 6) and 4 powerful divisors (1, 4, 9 and 36).
%t A360902 q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ e == 2^Length[e]]; q[1] = True; Select[Range[1300], q]
%o A360902 (PARI) is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, e[i]) == 2^#e; }
%Y A360902 Cf. A000005, A001694, A005117, A005361, A007947, A034444, A057521, A246655.
%Y A360902 Subsequence of A138302.
%Y A360902 Subsequences: A001248, A062503, A178739, A179747, A360903, A360904, A360905.
%K A360902 nonn
%O A360902 1,2
%A A360902 _Amiram Eldar_, Feb 25 2023