A387688 Number of solutions to n = 2^r + 3^s + 2^t * 3^u where r, s, t and u are nonnegative integers.

0, 0, 1, 2, 3, 5, 4, 5, 4, 4, 7, 6, 8, 7, 6, 4, 6, 5, 8, 7, 8, 4, 8, 0, 6, 5, 6, 4, 10, 3, 7, 5, 6, 7, 9, 6, 12, 6, 6, 4, 10, 3, 9, 7, 6, 4, 8, 0, 8, 2, 6, 4, 8, 0, 5, 2, 5, 3, 8, 3, 6, 3, 3, 2, 8, 2, 9, 5, 5, 2, 8, 0, 6, 4, 7, 4, 8, 0, 6, 0, 6, 3, 11, 3, 7, 6, 3, 3, 10, 3, 10, 7, 4, 5, 6, 0, 8, 6, 7

Offset

1,4

Comments

It is known that this sequence is bounded by a constant.

It appears that a(37) = 12 is the largest value to occur, a(83) = a(161) = 11 the second largest value(s), and for n > 299, all a(n) < 10. The values 7, 8, 9, 10 seem to occur for the last time at indices n = 2563, 2267, 515, 299; at least this is true up to 10^6. - M. F. Hasler, Sep 06 2025

Note that this sequence counts as solution each (distinct) tuple (r,s,t,u), while some authors count only distinct "representations" or sets {2^r, 3^s, 2^t*3^u}. Then, when t or u is zero, s and u (resp. r and t) become interchangeable, cf. Example. - M. F. Hasler, Sep 08 2025

Links

M. F. Hasler, Table of n, a(n) for n = 1..10000

P. Bajpai and M.A. Bennett, Effective S-unit Equations Beyond 3 Terms: Newman's Conjecture, Acta Arithmetica 214 (2024), 421-458, DOI:10.4064/aa230725-14-9; also arXiv:2308.05162.

Thomas Bloom, Problem 407, Erdős Problems.

J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, S-unit equations and their applications, New advances in transcendence theory (Durham, 1986) (1988), 138-140.

Example

For n = 4 the solutions are (0,0,1,0) and (1,0,0,0).
For n = 10 the solutions are (0,0,3,0), (0,1,1,1), (2,1,0,1), and (3,0,0,0).
For n = 531458, the a(n) = 5 solutions ((0,12,4,0), (3,2,0,12), (3,12,0,2), (4,0,0,12), (4,12,0,0)) correspond to only 2 distinct representations 1 + 2^4 + 3^12 and 2^3 + 3^2 + 3^12.

Mathematica

a[n_]:=Module[{p, q}, p=Ceiling[Log[2, n]]; q=Ceiling[Log[3, n]];  Sum[If[n==2^a+3^b+2^c*3^d, 1, 0], {a, 0, p}, {b, 0, q}, {c, 0, p}, {d, 0, q}]]; Table[a[n], {n, 1, 99}] (* James C. McMahon, Sep 14 2025 *)

Prog

(Python)
import math
def a(n):
    p, q = math.ceil(math.log2(n)), math.ceil(math.log(n, 3))
    return sum(1 for a in range(p+1) for b in range(q+1) for c in range(p+1) for d in range(q+1) if n == 2**a + 3**b + (2**c)*(3**d))
print([a(n) for n in range(1, 100)])
(PARI) A387688_upto(N)={my(A=[2^k|k<-[0..logint(N, 2)]], B=[3^k|k<-[0..logint(N, 3)]], S=vecsort(concat([[a+b|a<-A, a+b<N]|b<-B])), c=Vec(0, N)); foreach( vecsort(concat([[a*b|a<-A, a*b<N]|b<-B])), p, foreach(S, s, s+p>N && break; c[s+p]++)); c} \\ M. F. Hasler, Sep 06 2025

Crossrefs

Sequence in context: A102149 A321782 A104204 * A131296 A371216 A267808

Adjacent sequences: A387685 A387686 A387687 * A387689 A387690 A387691

Keyword

nonn

Author

Leon Grigoruk, Sep 06 2025

Status

approved