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A square array of numbers, read by antidiagonals, called Sundaram's sieve.
4

%I #42 Jul 27 2023 15:31:06

%S 4,7,7,10,12,10,13,17,17,13,16,22,24,22,16,19,27,31,31,27,19,22,32,38,

%T 40,38,32,22,25,37,45,49,49,45,37,25,28,42,52,58,60,58,52,42,28,31,47,

%U 59,67,71,71,67,59,47,31,34,52,66,76,82,84,82,76,66,52,34

%N A square array of numbers, read by antidiagonals, called Sundaram's sieve.

%C The sieve of Sundaram contains every number n > 3 for which the number 2*n + 1 is composite. For any n absent from this array, 2*n + 1 is a prime.

%D Ross Honsberger, Ingenuity in Mathematics, New Mathematical Library #23, Mathematical Association of America, 1970 (ISBN 0394709233); p. 75.

%D C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, Inc., New York, 1966.

%H Paolo Xausa, <a href="/A159919/b159919.txt">Table of n, a(n) for n = 1..11325</a> (antidiagonals 1..150 of the array, flattened)

%H Andrew Baxter, <a href="http://banach.millersville.edu/~bob/math478/History/Sundaram.html">Sundaram's Sieve</a>.

%H Julian Havil, <a href="http://plus.maths.org/issue50/features/havil/index.html">Sundaram's Sieve</a>, Plus Magazine, March 2009.

%H New Zealand Maths, <a href="https://web.archive.org/web/20040309001101/www.nzmaths.co.nz/HelpCentre/Newsletter18.pdf">Newletter 18</a>, October 2002.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sieve_of_Sundaram">Sundaram's Sieve</a>.

%F For the term in row j and column k, we have T[j, k] = 2*j*k + j + k.

%e For the term in row 3 and column 3, we have T[3, 3] = 2*3*3 + 3 + 3 = 24. Thus, 2*T[3,3] + 1 = 49 is composite.

%e From _Petros Hadjicostas_, Jun 19 2019: (Start)

%e The square array begins as follows:

%e 4, 7, 10, 13, 16, 19, ...

%e 7, 12, 17, 22, 27, ...

%e 10, 17, 24, 31, ...

%e 13, 22, 31, ...

%e 16, 27, ...

%e 19, ...

%e ...

%e (End)

%t A159919list[dmax_]:=Table[2k(j-k+1)+j+1,{j,dmax},{k,j}];A159919list[10] (* Generates 10 antidiagonals *) (* _Paolo Xausa_, Jul 26 2023 *)

%Y Cf. A159200, A159920.

%K easy,nonn,tabl

%O 1,1

%A _Russell Walsmith_, Apr 25 2009

%E More terms from _Philippe Deléham_, May 11 2009