A004050 Numbers of the form 2^j + 3^k, for j and k >= 0.

2, 3, 4, 5, 7, 9, 10, 11, 13, 17, 19, 25, 28, 29, 31, 33, 35, 41, 43, 59, 65, 67, 73, 82, 83, 85, 89, 91, 97, 113, 129, 131, 137, 145, 155, 209, 244, 245, 247, 251, 257, 259, 265, 275, 283, 307, 337, 371, 499, 513, 515, 521, 539, 593, 730, 731, 733, 737, 745, 755

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• Douglas Edward Iannucci, On duplicate representations as 2^x+3^y for nonnegative integers x and y, arXiv:1907.03347 [math.NT], 2019.

Crossrefs

Cf. A085634, A219835.

Cf. A226806-A226832 (cases to 8^j + 9^k).

Cf. A004051 (primes), A000079, A000243.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a004050 n = a004050_list !! (n-1)
a004050_list = f 1 $ singleton (2, 1, 1) where
   f x s = if y /= x then y : f y s'' else f x s''
           where s'' = insert (u * 2 + v, u * 2, v) $
                       insert (u + 3 * v, u, 3 * v) s'
                 ((y, u, v), s') = deleteFindMin s
-- Reinhard Zumkeller, May 20 2015

Maple

lincom:=proc(a,b,n) local i,j,s,m; s:={}; for i from 0 to n do for j from 0 to n do m:=a^i+b^j; if m<=n then s:={op(s),m} fi od; od; lprint(sort([op(s)])); end: lincom(2,3,760); # Zerinvary Lajos, Feb 24 2007

Mathematica

mx = 760; s = Union@ Flatten@ Table[2^i + 3^j, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx - 2^i]}] (* Robert G. Wilson v, Sep 19 2012 *)

PARI

ispow2(n)=n>>valuation(N,2)==1
is(n)=my(k); if(n%2, if(n<3, return(0)); for(k=0,logint(n-2,3), if(ispow2(n-3^k), return(1))); 0, ispower(n-1,,&k); k==3 || n==2 || n==4) \\ Charles R Greathouse IV, Aug 29 2016

Python

def aupto(lim):
    s, pow3 = set(), 1
    while pow3 < lim:
        for j in range((lim-pow3).bit_length()):
            s.add(2**j + pow3)
        pow3 *= 3
    return sorted(set(s))
print(aupto(756)) # Michael S. Branicky, Jul 29 2021

Formula

There are log^2 x/(log 2 log 3) + O(log x) terms up to x. Bounds on the error term can be made explicit. - Charles R Greathouse IV, Oct 28 2022

Extensions

More terms from Sascha Kurz, Jan 02 2003