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A372401
Position of 210^n among 7-smooth numbers A002473.
4
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.
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
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jun 03 2024
STATUS
approved