A004050 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

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

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 *)

PROG

(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

(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

CROSSREFS

Cf. A085634, A219835.

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