A159919 A square array of numbers, read by antidiagonals, called Sundaram's sieve.

4, 7, 7, 10, 12, 10, 13, 17, 17, 13, 16, 22, 24, 22, 16, 19, 27, 31, 31, 27, 19, 22, 32, 38, 40, 38, 32, 22, 25, 37, 45, 49, 49, 45, 37, 25, 28, 42, 52, 58, 60, 58, 52, 42, 28, 31, 47, 59, 67, 71, 71, 67, 59, 47, 31, 34, 52, 66, 76, 82, 84, 82, 76, 66, 52, 34

Offset

1,1

Comments

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.

References

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

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

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)

Andrew Baxter, Sundaram's Sieve.

Julian Havil, Sundaram's Sieve, Plus Magazine, March 2009.

New Zealand Maths, Newletter 18, October 2002.

Wikipedia, Sundaram's Sieve.

Formula

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

Example

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.
From Petros Hadjicostas, Jun 19 2019: (Start)
The square array begins as follows:
   4,  7, 10, 13, 16, 19, ...
   7, 12, 17, 22, 27, ...
  10, 17, 24, 31, ...
  13, 22, 31, ...
  16, 27, ...
  19, ...
  ...
(End)

Mathematica

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

Crossrefs

Cf. A159200, A159920.

Sequence in context: A199435 A257898 A333286 * A310929 A310930 A131432

Adjacent sequences: A159916 A159917 A159918 * A159920 A159921 A159922

Keyword

easy,nonn,tabl

Author

Russell Walsmith, Apr 25 2009

Extensions

More terms from Philippe Deléham, May 11 2009

Status

approved