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

1, 4, 9, 25, 36, 48, 49, 80, 100, 112, 121, 162, 169, 176, 196, 208, 225, 272, 289, 304, 361, 368, 405, 441, 464, 484, 496, 529, 567, 592, 656, 676, 688, 720, 752, 841, 848, 891, 900, 944, 961, 976, 1008, 1053, 1072, 1089, 1136, 1156, 1168, 1200, 1225, 1250, 1264

Offset: 1

Author

Amiram Eldar, Feb 25 2023

Keywords

• nonn

Comments

Numbers k such that A034444(k) = A005361(k).

Numbers whose squarefree kernel (A007947) and powerful part (A057521) have the same number of divisors (A000005).

If k and m are coprime terms, then k*m is also a term.

All the terms are exponentially 2^n-numbers (A138302).

The characteristic function of this sequence depends only on prime signature.

Numbers whose canonical prime factorization has exponents whose geometric mean is 2.

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).

1 is the only squarefree (A005117) term.

Includes the squares of squarefree numbers (A062503), which are the powerful (A001694) terms of this sequence.

The squares of primes (A001248) are the only terms that are prime powers (A246655).

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).

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).

Examples

			4 is a term since it has 2 squarefree divisors (1 and 2) and 2 powerful divisors (1 and 4).
36 is a term since it has 4 squarefree divisors (1, 2, 3 and 6) and 4 powerful divisors (1, 4, 9 and 36).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001694, A005117, A005361, A007947, A034444, A057521, A246655.

Subsequence of A138302.

Subsequences: A001248, A062503, A178739, A179747, A360903, A360904, A360905.

Programs

Mathematica

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ e == 2^Length[e]]; q[1] = True; Select[Range[1300], q]

PARI

is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, e[i]) == 2^#e; }