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
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
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jan 23 2010
STATUS
approved