A373559 Squares k such that rad(k) is a primorial number.

1, 4, 16, 36, 64, 144, 256, 324, 576, 900, 1024, 1296, 2304, 2916, 3600, 4096, 5184, 8100, 9216, 11664, 14400, 16384, 20736, 22500, 26244, 32400, 36864, 44100, 46656, 57600, 65536, 72900, 82944, 90000, 104976, 129600, 147456, 176400, 186624, 202500, 230400, 236196, 262144

Offset

1,2

Comments

Squares k such that the squarefree kernel of k is primorial.

Intersection of A000290 and A055932.

1 is the only primorial term.

From Michael De Vlieger, Jun 09 2024: (Start)

Contains k^2 for k in each of A000079, A033845, A143207, A147571, A147572, etc.

Contains k^2 such that k is a product of primorials, i.e., A025487(i)^2, i >= 1, since A025487 is a proper subset of A055932. (End)

Links

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

Formula

a(n) = A055932(n)^2.

Example

1 is a square, rad(1) = 1 = A002110(0).
4 is a square and rad(4) = 2 = A002110(1).
36 is a square and rad(36) = 6 = A002110(2).

Mathematica

{1}~Join~Select[Range[2, 512, 2], Or[# == {2}, Union@ Differences@ PrimePi[#] == {1}] &@ FactorInteger[#][[All, 1]] &]^2 (* Michael De Vlieger, Jun 09 2024 *)

Prog

(Python)
from functools import lru_cache
from itertools import count
from math import isqrt
from sympy import prime, integer_log
from oeis_sequences.OEISsequences import bisection
def A373559(n):
    @lru_cache(maxsize=None)
    def g(x, m): return sum(g(x//(prime(m)**i), m-1) for i in range(1, integer_log(x, prime(m))[0]+1)) if m-1 else x.bit_length()-1
    def f(x):
        c, p, y = n-1+x, 1, isqrt(x)
        for k in count(1):
            p *= prime(k)
            if p>y:
                break
            c -= g(y, k)
        return c
    return bisection(f, n, n) # Chai Wah Wu, Apr 02 2026
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import nextprime
def A373559_gen(): # generator of terms if the first n terms are desired.
    h, hset = [(1, (1, ))], {1}
    while True:
        m, ps = heappop(h)
        yield m
        for p in ps:
            mp = m*p**2
            if mp not in hset:
                heappush(h, (mp, ps))
                hset.add(mp)
        q = nextprime(max(ps, default=1))
        mp = m*q**2
        if mp not in hset:
            heappush(h, (mp, (ps+(q, ))))
            hset.add(mp)
A373559_list = list(islice(A373559_gen(), 43)) # Chai Wah Wu, Apr 02 2026

Crossrefs

Cf. A000290, A002110, A007947, A055932.

Sequence in context: A221285 A121317 A238259 * A063755 A166721 A085040

Adjacent sequences: A373556 A373557 A373558 * A373560 A373561 A373562

Keyword

nonn,easy

Author

David James Sycamore, Jun 09 2024

Extensions

More terms from David A. Corneth, Jun 09 2024

Status

approved