The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000952 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A000952

Numbers k == 2 (mod 4) that are the orders of conference matrices.
(history; published version)

#53 by Michael De Vlieger at Tue Jul 25 23:32:24 EDT 2023

STATUS

proposed

approved

#52 by Jon E. Schoenfield at Tue Jul 25 22:50:42 EDT 2023

STATUS

editing

proposed

#51 by Jon E. Schoenfield at Tue Jul 25 22:48:40 EDT 2023

NAME

Numbers n k == 2 (mod 4) that are the orders of conference matrices.

COMMENTS

A conference matrix of order n k is an n a k X n k {-1,0,+1} matrix A such that A A' = (nk-1)I.

If n k == 2 (mod 4) then a necessary condition is that nk-1 is a sum of 2 squares (A286636). It is conjectured that this condition is also sufficient. If n k == 2 (mod 4) and nk-1 is a prime or prime power the condition is automatically satisfied.

LINKS

Wikipedia, <a href="https://en.wikipedia.org/wiki/Conference_matrix">Conference matrix</a>.

CROSSREFS

CFCf. A286636.

EXTENSIONS

66 seems to be the smallest order for which it is not known if whether a conference matrix exists. Since 65 is the sum of two squares, according to the conjecture, 66 should be the next term.

STATUS

approved

editing

#50 by Michael De Vlieger at Sun Jan 01 12:36:15 EST 2023

STATUS

proposed

approved

#49 by Michel Marcus at Sun Jan 01 11:38:53 EST 2023

STATUS

editing

proposed

#48 by Michel Marcus at Sun Jan 01 11:38:42 EST 2023

CROSSREFS

Subsequence of A016825.

CF. A286636.

STATUS

approved

editing

#47 by Alois P. Heinz at Tue Aug 24 13:09:46 EDT 2021

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proposed

approved

#46 by Jon E. Schoenfield at Mon Aug 23 23:28:02 EDT 2021

STATUS

editing

proposed

#45 by Jon E. Schoenfield at Mon Aug 23 23:27:48 EDT 2021

COMMENTS

If n == 2 (mod 4) then a necessary condition is that n-1 is a sum of 2 squares (A286636). It is conjectured that this condition is also sufficient. If n == 2 (mod 4 ) and n-1 is a prime or prime power the condition is automatically satisfied.

STATUS

proposed

editing

#44 by Michel Marcus at Sun Aug 22 13:03:26 EDT 2021

STATUS

editing

proposed