A396778 - OEIS

%I #9 Jun 22 2026 21:19:51

%S 27,216,243,343,1331,1728,2187,2744,3375,6859,7776,9261,10648,12167,

%T 13824,16807,19683,21952,27000,29791,35937,42875,54872,59319,74088,

%U 79507,85184,97336,103823,110592,132651,157464,161051,166375,175616,177147,185193,205379,216000

%N Perfect powers not representable as sums of two squares.

%C Perfect powers m^k with odd k >= 3 such that m is not representable as a sum of two squares.

%C By Fermat's theorem on sums of two squares, this holds iff m contains a prime congruent to 3 mod 4 (A002145) with odd exponent.

%C Equivalently, these are perfect powers m^k with odd exponent k such that m is in A022544.

%C No squares occur in this sequence.

%C This sequence generalizes A125111, which corresponds to the case k = 3.

%H Felix Huber, <a href="/A396778/b396778.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares">Fermat's theorem on sums of two squares</a>.

%F a(n) are the numbers m^k with k >= 3 odd such that m is in A022544.

%e 216 = 6^3 is a term, since 6 contains the prime 3 == 3 (mod 4) to odd exponent; hence 216 is not representable as a sum of two squares.

%e 16807 = 7^5 is a term, since 7 is congruent to 3 mod 4 and occurs to odd exponent; hence 16807 is not representable as a sum of two squares.

%p A396778List := proc(N)

%p     local b, l, p, k, m, r, t;

%p     b := table(); l := floor(N^(1/3));

%p     for m from 2 to l do b[m] := false end do;

%p     for p from 3 to l do

%p         if isprime(p) and p mod 4 = 3 then

%p             for k from 1 to floor(log(l)/log(p)) do

%p                 if k mod 2 = 1 then

%p                     for m from p^k to l by p^k do

%p                         if irem(m, p^(k + 1)) <> 0 then b[m] := true end if

%p                     end do

%p                 end if

%p             end do

%p         end if

%p     end do;

%p     t := table();

%p     for m from 2 to l do

%p         if b[m] then

%p             r := m^3;

%p             while r <= N do

%p                 t[r] := true; r := r*m^2

%p             end do

%p         end if

%p     end do;

%p     return sort([indices(t, 'nolist')])

%p end proc:

%p A396778List(216000);

%t With[{nn = 2^18}, Union@ Flatten[Table[If[SquaresR[2, #] == 0, #, Nothing] &[k^m], {m, 2, Log2[nn]}, {k, 2, Surd[nn, m]} ] ] ] (* _Michael De Vlieger_, Jun 18 2026 *)

%Y Subsequence of A001597.

%Y Supersequence of A125111.

%Y Disjoint from A000290.

%Y Cf. A001481, A002145, A022544.

%K nonn,easy,new

%O 1,1

%A _Felix Huber_, Jun 18 2026