cp's OEIS Frontend

A306044 Powers of 2, 3 and 5.

%I A306044 #46 Feb 05 2025 12:30:53
%S A306044 1,2,3,4,5,8,9,16,25,27,32,64,81,125,128,243,256,512,625,729,1024,
%T A306044 2048,2187,3125,4096,6561,8192,15625,16384,19683,32768,59049,65536,
%U A306044 78125,131072,177147,262144,390625,524288,531441,1048576,1594323,1953125,2097152,4194304,4782969,8388608
%N A306044 Powers of 2, 3 and 5.
%C A306044 Union of A000079, A000244 and A000351.
%H A306044 Michel Marcus, <a href="/A306044/b306044.txt">Table of n, a(n) for n = 1..1370</a>
%F A306044 Sum_{n>=1} 1/a(n) = 11/4. - _Amiram Eldar_, Dec 10 2022
%p A306044 N:= 10^7: # for terms <= N
%p A306044 sort(convert(`union`(seq({seq(b^i,i=0..ilog[b](N))},b=[2,3,5])),list)); # _Robert Israel_, Nov 18 2022
%t A306044 Union[2^Range[0, Log2[5^10]], 3^Range[Log[3, 5^10]], 5^Range[10]]
%o A306044 (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
%o A306044 (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).
%o A306044 (Python)
%o A306044 from sympy import integer_log
%o A306044 def A306044(n):
%o A306044     def bisection(f,kmin=0,kmax=1):
%o A306044         while f(kmax) > kmax: kmax <<= 1
%o A306044         kmin = kmax >> 1
%o A306044         while kmax-kmin > 1:
%o A306044             kmid = kmax+kmin>>1
%o A306044             if f(kmid) <= kmid:
%o A306044                 kmax = kmid
%o A306044             else:
%o A306044                 kmin = kmid
%o A306044         return kmax
%o A306044     def f(x): return n+x-x.bit_length()-integer_log(x,3)[0]-integer_log(x,5)[0]
%o A306044     return bisection(f,n,n) # _Chai Wah Wu_, Feb 05 2025
%Y A306044 Cf. A000079, A000244, A000351, A006899.
%Y A306044 Cf. A226722, A226723, A226724.
%K A306044 nonn
%O A306044 1,2
%A A306044 _Zak Seidov_, Jun 18 2018