A352598 -id:A352598 - cp's OEIS frontend

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

Michael S. Branicky, Mar 24 2022

nonn

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.

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

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

Intersection of A000290 and A002473.

Cf. A352598.

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 *)

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)))

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

# 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

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

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

A352595 Positive integers that are fixed points for the map x->f^k(x) for some k>1, where f(x) is the product of squares of nonzero digits of x.

Original entry on oeis.org

256, 324, 576, 3600, 11664, 15876, 20736, 44100, 63504, 65536, 129600, 2822400, 5308416, 7290000, 8294400

Offset: 1

Michael S. Branicky, Mar 24 2022

f(x) = A352598(x).
Fixed points of f(x) are in A115385.
64524128256, 386983526400, 849346560000, 49787136000000, 55725627801600 are also terms.

			256 -> 3600 -> 324 -> 576 -> 44100 -> 256 is a limit cycle of f, so all elements are terms.

Subsequence of the intersection of A000290 and A002473.

Cf. A115385, A351327, A352598.

from math import prod
from itertools import count, islice
def f(n): return prod(int(d)**2 for d in str(n) if d != "0")
def ok(n):
    n0, k, seen = n, 0, set(),
    while n not in seen: # iterate until fixed point or in cycle
        seen.add(n)
        n = f(n)
        k += 1
    return n == n0 and k > 1
def agen(startk=1):
    for m in count(1):
        if ok(m): yield m
print(list(islice(agen(), 11)))

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A352618 Squares that are 7-smooth.

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A352595 Positive integers that are fixed points for the map x->f^k(x) for some k>1, where f(x) is the product of squares of nonzero digits of x.

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Python