A376102 Array read by ascending antidiagonals: A(n,k) = k*2^(n+1) + 1.

1, 1, 3, 1, 5, 5, 1, 9, 9, 7, 1, 17, 17, 13, 9, 1, 33, 33, 25, 17, 11, 1, 65, 65, 49, 33, 21, 13, 1, 129, 129, 97, 65, 41, 25, 15, 1, 257, 257, 193, 129, 81, 49, 29, 17, 1, 513, 513, 385, 257, 161, 97, 57, 33, 19, 1, 1025, 1025, 769, 513, 321, 193, 113, 65, 37, 21

Offset: 0

Author

Stefano Spezia, Sep 14 2024

Keywords

• nonn
• easy
• tabl

Comments

In 1747, Euler showed that any factor of a Fermat number A000215(n) is of the form k*2^(n+1) + 1. See Wells at p. 148.

Examples

			The array begins as:
  1,   3,   5,   7,   9,  11,  13, ...
  1,   5,   9,  13,  17,  21,  25, ...
  1,   9,  17,  25,  33,  41,  49, ...
  1,  17,  33,  49,  65,  81,  97, ...
  1,  33,  65,  97, 129, 161, 193, ...
  1,  65, 129, 193, 257, 321, 385, ...
  1, 129, 257, 385, 513, 641, 769, ...
  ...
		

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 70-71, 237-242.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 136.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987.

Crossrefs

Cf. A000079, A000215, A000295.

Cf. A000012 (k=0), A000051, A000337, A004119, A005408 (n=0), A016813 (n=1), A017077 (n=2), A158057 (n=3).

Programs

Mathematica

A[n_,k_]:=k*2^(n+1)+1; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

G.f.: (1 - 2*x + y)/((1 - x)*(1 - 2*x)*(1 - y)^2).

E.g.f.: exp(x+y)*(1 + 2*exp(x)*y).

Sum_{0<=k<=n} A(n-k,k) = A000295(n+2).

A(n,1) = A000051(n+1).

A(n,3) = A004119(n+2).

A(n,n) = A000337(n+1).