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%I #8 Jul 07 2018 19:26:18
%S 5,7,11,13,17,19,23,29,31,37,43,47,53,59,61,67,71,73,83,89,97,101,103,
%T 107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,
%U 197,199,211,223,227,229,233,251,257,163,271,277,181,283,293,307,311
%N Petoukhov primes, generated by M*H*M; M = 2^n circulant matrices generated from A164281, H = all inequivalent Hadamard matrices of order 2^n.
%C The basic idea for the sequence was conceived by Sergey Petoukhov; coupled with the strategy of using circulant matrices in M.
%H N. J. A. Sloane, <a href="http://neilsloane.com/hadamard/">Tables of Hadamard Matrices</a>
%F 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 (+).
%e 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.
%e The product M*H*M =
%e ...
%e -7, 1, 29, 13;
%e 1, 23, 37, 11;
%e 29, 37, 47, 31;
%e 13, 11, 13, 17;
%e ... Then extract all terms that are primes, becoming the ordered set, A172039.
%e 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).
%e 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.
%Y Cf. A000040, A164281.
%K nonn
%O 1,1
%A _Gary W. Adamson_, Jan 23 2010