A172039 Petoukhov primes, generated by M*H*M; M = 2^n circulant matrices generated from A164281, H = all inequivalent Hadamard matrices of order 2^n.

5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 251, 257, 163, 271, 277, 181, 283, 293, 307, 311

Offset

1,1

Comments

The basic idea for the sequence was conceived by Sergey Petoukhov; coupled with the strategy of using circulant matrices in M.

Links

Table of n, a(n) for n=1..57.

N. J. A. Sloane, Tables of Hadamard Matrices

Formula

Let M = 2^n x 2^n circulant matrices generated from rows of A164281: (1; 1,2; 1,2,4,2; 1,2,4,2,4,8,4,2;...) and H = inequivalent Hadamard matrices of order 2^n. A172039 consists of the primes extracted from the products M*H*M using all of the Hadamard matrices in orders 2^n. Last, change and (-) signs to (+).

Example

The 4 X 4 circulant matrix using A164281 = [1,2,4,2; 2,1,2,4; 4,2,1,2; 2,4,2,1] = M. The 4 X 4 inequivalent Hadamard matrix = [ ++++; +-+-; ++--; +--+ ] = H.
The product M*H*M =
  ...
  -7,  1, 29, 13;
   1, 23, 37, 11;
  29, 37, 47, 31;
  13, 11, 13, 17;
  ... Then extract all terms that are primes, becoming the ordered set, A172039.
Similarly, with order 16 we create a 16 X 16 circulant matrix M using the terms (1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2), (Cf. A164281).
Using (16.4 Hadamard matrix = H; Cf. Tables of Hadamard Matrices); we take the product M*H*M, extracting the primes and putting them into the ordered set. The top row of that product = (487, 95, 197, 637, 31, 241, 1085, 109, 355, 227, 55, 313, 31, 97, 341, 443), with the primes: 487, 197, 31, 241, 109, 227, 313, 31, 97, and 443.

Crossrefs

Cf. A000040, A164281.

Sequence in context: A020630 A216775 A020586 * A020602 A020617 A299760

Adjacent sequences: A172036 A172037 A172038 * A172040 A172041 A172042

Keyword

nonn

Author

Gary W. Adamson, Jan 23 2010

Status

approved