A391320 Squares of squarefree numbers that have at least 3 distinct prime factors.

900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27225, 28900, 30276, 33124, 34596, 36100, 38025, 44100, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 74529, 79524, 81225, 81796, 84100, 96100, 101124, 103684, 108900, 119025

Offset

1,1

Comments

Squares k^2 of k in A350352 = A177492 \ A085986.

Intersection of A000977, A004709, and A001597.

Superset of A162143; a(20) = 44100 is not in A162143 since 44100 has 4 distinct prime factors.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Formula

Sum_{n>=1} 1/a(n) = 15/Pi^2 - (P(2)^2 - P(4))/2 - P(2) - 1 = 0.0038030400092245298715..., where P(k) is the prime zeta function. - Amiram Eldar, Dec 10 2025

Example

Table of n, a(n) for select n:
 n     a(n)
-------------------------------------------
 1     900 =  30^2 = 2^2 * 3^2 *  5^2
 2    1764 =  42^2 = 2^2 * 3^2 *  7^2
 3    4356 =  60^2 = 2^2 * 3^2 * 11^2
 4    4900 =  70^2 = 2^2 * 5^2 *  7^2
 5    6084 =  78^2 = 2^2 * 3^2 * 13^2
 6   10404 = 102^2 = 2^2 * 3^2 * 17^2
 7   11025 = 105^2 = 3^2 * 5^2 *  7^2
 8   12100 = 110^2 = 2^2 * 5^2 * 11^2
 9   12996 = 114^2 = 2^2 * 3^2 * 19^2
10   16900 = 130^2 = 2^2 * 5^2 * 13^2
11   19044 = 138^2 = 2^2 * 3^2 * 23^2
20   44100 = 210^2 = 2^2 * 3^2 *  5^2 * 7^2

Mathematica

Select[Range[360], And[SquareFreeQ[#], PrimeNu[#] > 2] &]^2

Prog

(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
from oeis_sequences.OEISsequences import bisection
def A391320(n):
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b+1, isqrt(x//c)+1), a+1)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b+1, integer_nthroot(x//c, m)[0]+1), a+1) for d in g(x, a2, b2, c*b2, m-1)))
    def f(x): return int(n+x-sum(sum(primepi(isqrt(x)//prod(c[1] for c in a))-a[-1][0] for a in g(isqrt(x), 0, 1, 1, i)) for i in range(3, x.bit_length()>>1)))
    return bisection(f, n, n) # Chai Wah Wu, Dec 10 2025

Crossrefs

Cf. A000977, A001597, A001694, A005117, A085986, A126706, A131605, A162143, A177492, A286708, A303606, A350352.

Sequence in context: A391427 A074853 A392741 * A162143 A391428 A258888

Adjacent sequences: A391317 A391318 A391319 * A391321 A391322 A391323

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 08 2025

Status

approved