cp's OEIS Frontend

A386373 a(n) is the smallest integer w such that the equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly n positive integer solutions.

%I A386373 #31 Aug 11 2025 10:31:44
%S A386373 2,7,17,25,9,100,44,57,117,49,73,81,33,89,177,193,305,161,257,273,425,
%T A386373 289,697,441,313,689,369,593,809,233,761,1865,2001,857,1121,649,1353,
%U A386373 865,521,1257,577,681,2081,1409,1169,1753,1801,1201,1745,2833,3853,3649,3353,1305,793
%N A386373 a(n) is the smallest integer w such that the equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly n positive integer solutions.
%C A386373 From _David A. Corneth_, Jul 20 2025: (Start)
%C A386373 a(41) = 577. If a(41) is 1 (mod 8) then that values is exact.
%C A386373 For 10 <= n <= 30 we have a(n) == 1 (mod 8).
%C A386373 Heuristically this is no coincidence. There are 8^3 = 512 tuples (x, y, z) mod 8. The frequencies of k (mod 8) for x^2 + y^3 + z^4 for k = 0 through 7 are 64, 128, 96, 32, 64, 64, 32, 32 respectively. So 1 (mod 8) has the single largest value at 128 such tuples.
%C A386373 Extending this to other moduli like 56 we get the largest frequencies (7168) come from 9, 17, 25 and 33 (mod 56).
%C A386373 The second largest frequency is 6272 which occurs at 49 (mod 56). For n = 3, 4 and 10 <= n <= 20, 22, 30 we have a(n) == 9, 17, 25, 33 or 49 (mod 56). (End)
%H A386373 Zhining Yang, <a href="/A386373/b386373.txt">Table of n, a(n) for n = 1..70</a>
%e A386373 a(4) = 25 because 25^5 = 1852^2 + 185^3 + 8^4 = 2711^2 + 134^3 + 10^4 = 2472^2 + 150^3 + 23^4 = 2973^2 + 15^3 + 31^4 and no integer less than 25 has 4 solutions.
%t A386373 f[w_]:=(v={};c=0;nn=w^5;
%t A386373 Do[yy=nn-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
%t A386373 If[IntegerQ@x,If[GCD[x,y,z]==1,AppendTo[v,{x,y,z,d}];c++]],{y,Floor[yy^(1/3)]}],{z,Floor[nn^(1/4)]}];{c,w,v});
%t A386373 s=Table[{},20];
%t A386373 For[k=1,k<=100,k++,r=f[k][[1]];If[s[[r]]=={},s[[r]]=g[k];Print[s[[r]]]]]
%Y A386373 Cf. A386377, A386521.
%K A386373 nonn
%O A386373 1,1
%A A386373 _Zhining Yang_, Jul 19 2025
%E A386373 a(21)-a(31) from _David A. Corneth_, Jul 20 2025
%E A386373 a(32)-a(58) from _Zhining Yang_, Jul 31 2025