A352618 Squares that are 7-smooth.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 144, 196, 225, 256, 324, 400, 441, 576, 625, 729, 784, 900, 1024, 1225, 1296, 1600, 1764, 2025, 2304, 2401, 2500, 2916, 3136, 3600, 3969, 4096, 4900, 5184, 5625, 6400, 6561, 7056, 8100, 9216, 9604, 10000, 11025, 11664, 12544, 14400

Offset: 1

Author

Michael S. Branicky, Mar 24 2022

Keywords

• nonn

Comments

Also, distinct terms appearing in A352598, or terms of the form 4^i * 9^j * 25^k * 49^m for i, j, k, m >= 0.

Examples

			49 = 7*7, 81 = (3*3)*(3*3), and 100 = (2*5)*(2*5) are terms.
		

Links

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

Crossrefs

Intersection of A000290 and A002473.

Cf. A352598.

Programs

Mathematica

Select[Range[120], Max[FactorInteger[#][[;; , 1]]] <= 7 &]^2 (* Amiram Eldar, Mar 24 2022 *)
With[{n = 15000}, Union@ Flatten@ Table[2^(2 a)*3^(2 b)*5^(2 c)*7^(2 d), {a, 0, Log[4, n]}, {b, 0, Log[9, n/(2^(2 a))]}, {c, 0, Log[25, n/(2^(2 a)*3^(2 b))]}, {d, 0, Log[49, n/(2^(2 a)*3^(2 b)*5^(2 c))]}]] (* Michael De Vlieger, Mar 26 2022 *)

Python

from itertools import count, islice
def agen():
    for i in count(1):
        k = i
        for p in [2, 3, 5, 7]:
            while k%p == 0:
                k //= p
        if k == 1:
            yield i*i
print(list(islice(agen(), 50)))

Python

from sympy import integer_log
def A352618(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    return bisection(f,n,n)**2 # Chai Wah Wu, Sep 17 2024

Python

# faster for initial segment of sequence
import heapq
from itertools import islice
from sympy import primerange
def A352618gen(p=7): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v*v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A352618gen(), 65))) # Michael S. Branicky, Sep 17 2024

Formula

a(n) = A002473(n)^2. - Pontus von Brömssen, Mar 24 2022

Sum_{n>=1} 1/a(n) = 1225/768. - Amiram Eldar, Mar 24 2022