A085634 -id:A085634 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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.

Cf. A085634, A219835.
Cf. A226806-A226832 (cases to 8^j + 9^k).
Cf. A004051 (primes), A000079, A000243.

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

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

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 *)

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

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

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

More terms from Sascha Kurz, Jan 02 2003

A141747 a(n) is the number of nonnegative integer pairs i,j such that n = 2^i + 3^j.

Original entry on oeis.org

0, 1, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Jul 01 2008

nonn

			a(5) = 2, using (i,j) = (1,1) and (2,0).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A085634, A141748.

f:= proc(n) local j,t,s;
  t:= 0;
  for j from 0 to floor(log[3](n)) do
    s:= n - 3^j;
    if s = 2^padic:-ordp(s,2) then t:= t+1 fi
  od;
  t
end proc:
map(f, [$1..200]); # Robert Israel, Jun 08 2020

G.f.: ( Sum_{i>=0} x^(2^i) ) * ( Sum_{j>=0} x^(3^j) ). - Ilya Gutkovskiy, Feb 10 2022

A141748 a(n) = n-th nonnegative integer k such that there is a unique nonnegative integer pair (i,j) for which n = 2^i + 3^j.

Original entry on oeis.org

2, 3, 4, 7, 9, 10, 13, 19, 25, 28, 29, 31, 33, 41, 43, 59, 65, 67, 73, 82, 83, 85, 89, 91, 97, 113, 129, 131, 137, 145, 155, 209, 244, 245, 247, 251, 257, 265, 275, 283, 307, 337, 371, 499, 513, 515, 521, 539, 593, 730, 731, 733, 737, 745, 755, 761, 793, 857, 985

Offset: 1

Clark Kimberling, Jul 01 2008

nonn

Locations of 1's in A141747.

Cf. A141747, A085634.

cp's OEIS Frontend

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

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A141747 a(n) is the number of nonnegative integer pairs i,j such that n = 2^i + 3^j.

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A141748 a(n) = n-th nonnegative integer k such that there is a unique nonnegative integer pair (i,j) for which n = 2^i + 3^j.

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