A158106 A triangle of structure called "Polynomial on residue classes" (PORC).

1, 2, 2, 5, 5, 5, 14, 14, 15, 15, 51, 51, 67, 76, 77, 267, 267, 504, 633, 684, 731, 2328, 2328, 9310, 9310, 34297, 62440, 113147

Offset: 1

Author

Roger L. Bagula, Mar 12 2009

Keywords

• nonn
• tabl
• uned

Examples

			Triangle begins:
  {1},
  {2, 2},
  {5, 5, 5},
  {14, 14, 15, 15},
  {51, 51, 67, 76, 77},
  {267, 267, 504, 633, 684, 731},
  {2328, 2328, 9310, 9310, 34297, 62440, 113147}
		

References

• Marcus Du Sautoy, Symmetry: A Journey into the Patterns of Nature,Harper (March 11, 2008),page 96,pp. 141-143 ( and the work of Christopher Voll)

Links

• Brett Edward Witty, Enumeration of groups of prime-power order, PhD Thesis, 2006 (see p. 3, 4).

Programs

Mathematica

p[x_, 1] := 1;
p[x_, 2] := 2;
p[x_, 3] := 5;
p[x_, 4] := If[x <= 2, 14, 15];
p[x_, 5] := If[x <= 2, 51, If[x == 3, 67, 61 + 2*x + 2 * GCD[x - 1, 3] + GCD[x - 1, 4]]];
p[x_, 6] := If[x <= 2, 267, If[x == 3, 504, 3*x^2 + 39*x + 344 + 24 *GCD[x - 1, 3] + 11* GCD[x - 1, 4] + 2* GCD[x - 1, 5]]];
p[x_, 7] := If[x <= 2, 2328, If[x >= 3 and x<5, 9310, If[x == 5, 34297,
3*x^5 + 12*x^4 + 44*x^3 + 170*x^2 + 707* x + 2455 + (4*x^2 + 44*x + 291) *GCD[x - 1, 3] + (x^2 + 19*x + 135) * GCD[x - 1, 4] + (3*x + 31)* GCD[x - 1, 5] + 4* GCD[ x - 1, 7] + 5* GCD[x - 1, 8] + GCD[x - 1, 9]]]];
a = Table[Table[p[x, n], {x, 1, n}], {n, 1, 7}];
Flatten[a]