A367416 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.

4, 8, 1, 16, 1, 32, 0, 2, 64, 6, 128, 8, 256, 16, 4, 512, 26, 1024, 17, 10, 2048, 67, 4, 3, 4096, 100, 10, 8192, 137, 34, 6, 16384, 426, 28, 1, 32768, 661, 96, 6, 65536, 1351, 146, 16, 8, 131072, 2637, 230, 15, 262144, 3831, 258, 40, 524288, 8095, 1130, 50

Offset: 2

Author

Jean-Marc Rebert, Jan 26 2024

Keywords

• nonn
• tabf

Comments

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.

Examples

			Triangle begins:
            k = 1      2     3   4  5
  n= 2:         4;
  n= 3:         8,     1;
  n= 4:        16,     1;
  n= 5:        32,     0,    2;
  n= 6:        64,     6;
  n= 7:       128,     8;
  n= 8:       256,    16,    4;
  n= 9:       512,    26;
  n=10:      1024,    17,   10;
  n=11:      2048,    67,    4,  3;
  n=12:      4096,   100,   10;
  n=13:      8192,   137,   34,  6;
  n=14:     16384,   426,   28,  1;
  n=15:     32768,   661,   96,  6;
  n=16:     65536,  1351,  146, 16, 8;
  n=17:    131072,  2637,  230, 15;
  n=18:    262144,  3831,  258, 40;
  n=19:    524288,  8095, 1130, 50;
  n=20:   1048576, 15241,  854, 77, 6;
  ...
The T(6,2) = 6 solutions are:
  - 1^2 - 2^2 + 3^2 - 4^2 + 5^2 + 6^2 = 49 = 7^2,
  - 1^2 - 2^2 + 3^2 + 4^2 + 5^2 - 6^2 =  9 = 3^2,
  - 1^2 - 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 81 = 9^2,
  + 1^2 - 2^2 + 3^2 - 4^2 - 5^2 + 6^2 =  1 = 1^2,
  + 1^2 + 2^2 - 3^2 + 4^2 + 5^2 - 6^2 =  1 = 1^2,
  + 1^2 + 2^2 + 3^2 - 4^2 - 5^2 + 6^2 =  9 = 3^2.
		

Crossrefs

Cf. A063890, A215083, A368243, A368845, A369629.

Programs

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~
is(k,u)=my(x=f(k,u));ispower(x,k)
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