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

Author

Zak Seidov, Jun 18 2018

Keywords

• nonn

Comments

Union of A000079, A000244 and A000351.

Links

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

Crossrefs

Cf. A000079, A000244, A000351, A006899.

Cf. A226722, A226723, A226724.

Programs

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]]

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(jRuud 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

Formula

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