A388076 Integers which can be expressed as the sum of three distinct perfect powers.

13, 14, 18, 21, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90

Offset

1,1

Comments

The first integer which can be represented in two ways is 21, the first in three ways is 37, the first in four ways is 49, the first in five ways is 84. Multiple representations appear to become more prevalent as integers become larger.

Links

Table of n, a(n) for n=1..65.

Example

21 = 1+2^2+2^4 = 2^2+2^3+3^2.
37 = 1+2^2+2^5 = 1+3^2+3^3 = 2^2+2^3+5^2.
49 = 1+2^4+2^5 = 2^2+3^2+6^2 = 2^3+3^2+2^5 = 2^3+2^4+5^2.

Maple

N:= 1000: # for terms <= N
PP:= sort(convert({1, seq(seq(a^k, a=2..floor(N^(1/k))), k=2..floor(log[2](N)))}, list)):
nPP:= nops(PP):
filter:= proc(n) local i, j;
       for i from 1 to nPP-1 while PP[i] < n/3 do
         for j from i+1 to nPP-1 while PP[i] + 2*PP[j] < n do
           if member(n-PP[i]-PP[j], PP) then return true fi
      od od;
     false
end proc:
select(filter, [$1..N]); # Robert Israel, Dec 02 2025

Mathematica

With[{m = 90}, Select[Range[m], Count[IntegerPartitions[#, {3}, Select[Range[m], # == 1 || GCD @@ FactorInteger[#][[;; , 2]] > 1 &]], _?(UnsameQ @@ #1 &)] > 0 &]] (* Amiram Eldar, Nov 29 2025 *)

Crossrefs

Cf. A000419, A001597, A003072, A004432, A024975, A074499, A351066.

Complement of A135578.

Sequence in context: A173036 A241749 A385364 * A098045 A293817 A360097

Adjacent sequences: A388073 A388074 A388075 * A388077 A388078 A388079

Keyword

easy,nonn

Author

Paul Duckett, Nov 14 2025

Status

approved