A372401 Position of 210^n among 7-smooth numbers A002473.

1, 68, 547, 2119, 5817, 13008, 25412, 45078, 74409, 116147, 173379, 249532, 348375, 474018, 630922, 823885, 1058051, 1338898, 1672260, 2064302, 2521535, 3050825, 3659361, 4354687, 5144682, 6037582, 7041946, 8166692, 9421074, 10814695, 12357491, 14059744, 15932086, 17985473

Offset

0,2

Comments

Also position of 210^(n+1) in A147571.

Links

Table of n, a(n) for n=0..33.

Formula

a(n) ~ c * n^4, where c = log(210)^4/(24*log(2)*log(3)*log(5)*log(7)) = 14.282278766622... - Vaclav Kotesovec and Amiram Eldar, Sep 22 2024

Mathematica

Table[
  Sum[Floor@ Log[7, 210^n/(2^i*3^j*5^k)] + 1,
    {i, 0, Log[2, 210^n]},
    {j, 0, Log[3, 210^n/2^i]},
    {k, 0, Log[5, 210^n/(2^i*3^j)]}],
  {n, 0, 12}]

Prog

(Python)
import heapq
from itertools import islice
from sympy import primerange
def A372401gen(p=7): # generator for p-smooth terms
    v, oldv, psmooth_primes, = 1, 0, list(primerange(1, p+1))
    h = [(1, [0]*len(psmooth_primes))]
    idx = {psmooth_primes[i]:i for i in range(len(psmooth_primes))}
    loc = 0
    while True:
        v, e = heapq.heappop(h)
        if v != oldv:
            loc += 1
            if len(set(e)) == 1:
                yield loc
            oldv = v
            for p in psmooth_primes:
                vp, ep = v*p, e[:]
                ep[idx[p]] += 1
                heapq.heappush(h, (v*p, ep))
print(list(islice(A372401gen(), 15))) # Michael S. Branicky, Jun 05 2024
(Python)
from sympy import integer_log
def A372401(n):
    c, x = 0, 210**n
    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 # Chai Wah Wu, Sep 16 2024

Crossrefs

Cf. A002110, A002473, A022330, A147571, A202821, A372400, A372402.

Sequence in context: A032510 A211691 A027850 * A248467 A281565 A281883

Adjacent sequences: A372398 A372399 A372400 * A372402 A372403 A372404

Keyword

nonn

Author

Michael De Vlieger, Jun 03 2024

Status

approved