A306044 Powers of 2, 3 and 5.

1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 3125, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 8388608

Offset

1,2

Comments

Union of A000079, A000244 and A000351.

Links

Michel Marcus, Table of n, a(n) for n = 1..1370

Formula

Sum_{n>=1} 1/a(n) = 11/4. - Amiram Eldar, Dec 10 2022

Maple

N:= 10^7: # for terms <= N
sort(convert(`union`(seq({seq(b^i, i=0..ilog[b](N))}, b=[2, 3, 5])), list)); # Robert Israel, Nov 18 2022

Mathematica

Union[2^Range[0, Log2[5^10]], 3^Range[Log[3, 5^10]], 5^Range[10]]

Prog

(PARI) setunion(setunion(vector(logint(N=10^6, 5)+1, k, 5^(k-1)), vector(logint(N, 3), k, 3^k)), vector(logint(N, 2), k, 2^k)) \\ M. F. Hasler, Jun 24 2018
(PARI) a(n)= my(f=[2, 3, 5], q=sum(k=1, #f, 1/log(f[k]))); for(i=1, #f, my(p=logint(exp(n/q), f[i]), d=0, j=0, m=0); while(j<n, m=f[i]^(p+d); j=1+sum(k=1, #f, logint(m, f[k])); if(j==n, return(m)); d++)) \\ Ruud H.G. van Tol, Nov 16 2022 (with the help of the pari-users mailing list) Observation: with f=primes(P), d <= logint(P, 2).
(Python)
from sympy import integer_log
def A306044(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-x.bit_length()-integer_log(x, 3)[0]-integer_log(x, 5)[0]
    return bisection(f, n, n) # Chai Wah Wu, Feb 05 2025

Crossrefs

Cf. A000079, A000244, A000351, A006899.

Cf. A226722, A226723, A226724.

Sequence in context: A286431 A015931 A330400 * A263581 A120430 A295033

Adjacent sequences: A306041 A306042 A306043 * A306045 A306046 A306047

Keyword

nonn

Author

Zak Seidov, Jun 18 2018

Status

approved