Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #48 Jan 30 2024 02:54:22
%S 4,8,1,16,1,32,0,2,64,6,128,8,256,16,4,512,26,1024,17,10,2048,67,4,3,
%T 4096,100,10,8192,137,34,6,16384,426,28,1,32768,661,96,6,65536,1351,
%U 146,16,8,131072,2637,230,15,262144,3831,258,40,524288,8095,1130,50
%N Triangle read by rows: T(n,k) = number of solutions to +- 1^k +- 2^k +- 3^k +- ... +- n^k is a k-th power, n >= 2.
%C In the case of n = 1, there are solutions for all k. In particular, 1^k is always a k-th power and -(1^k) is a k-th power for odd k. As a formula: T(1,k) = 1 + (k mod 2). This row is not included in the sequence.
%e Triangle begins:
%e k = 1 2 3 4 5
%e n= 2: 4;
%e n= 3: 8, 1;
%e n= 4: 16, 1;
%e n= 5: 32, 0, 2;
%e n= 6: 64, 6;
%e n= 7: 128, 8;
%e n= 8: 256, 16, 4;
%e n= 9: 512, 26;
%e n=10: 1024, 17, 10;
%e n=11: 2048, 67, 4, 3;
%e n=12: 4096, 100, 10;
%e n=13: 8192, 137, 34, 6;
%e n=14: 16384, 426, 28, 1;
%e n=15: 32768, 661, 96, 6;
%e n=16: 65536, 1351, 146, 16, 8;
%e n=17: 131072, 2637, 230, 15;
%e n=18: 262144, 3831, 258, 40;
%e n=19: 524288, 8095, 1130, 50;
%e n=20: 1048576, 15241, 854, 77, 6;
%e ...
%e The T(6,2) = 6 solutions are:
%e - 1^2 - 2^2 + 3^2 - 4^2 + 5^2 + 6^2 = 49 = 7^2,
%e - 1^2 - 2^2 + 3^2 + 4^2 + 5^2 - 6^2 = 9 = 3^2,
%e - 1^2 - 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 81 = 9^2,
%e + 1^2 - 2^2 + 3^2 - 4^2 - 5^2 + 6^2 = 1 = 1^2,
%e + 1^2 + 2^2 - 3^2 + 4^2 + 5^2 - 6^2 = 1 = 1^2,
%e + 1^2 + 2^2 + 3^2 - 4^2 - 5^2 + 6^2 = 9 = 3^2.
%o (PARI)f(k,u)=my(x=0,v=vector(#u));for(i=1,#u,u[i]=if(u[i]==0,-1,1);v[i]=i^k);u*v~
%o is(k,u)=my(x=f(k,u));ispower(x,k)
%o T(n,k)=my(u=vector(n,i,[0,1]),nbsol=0);if(k%2==1,u[1]=[1,1]);forvec(X=u,if(is(k,X),nbsol++));if(k%2==1,nbsol*=2);nbsol
%Y Cf. A063890, A215083, A368243, A368845, A369629.
%K nonn,tabf
%O 2,1
%A _Jean-Marc Rebert_, Jan 26 2024