A154778 Numbers of the form a^2 + 5b^2 with positive integers a,b.

6, 9, 14, 21, 24, 29, 30, 36, 41, 45, 46, 49, 54, 56, 61, 69, 70, 81, 84, 86, 89, 94, 96, 101, 105, 109, 116, 120, 126, 129, 134, 141, 144, 145, 149, 150, 161, 164, 166, 174, 180, 181, 184, 189, 196, 201, 205, 206, 214, 216, 224, 225, 229, 230, 241, 244, 245, 246

Offset

1,1

Comments

Subsequence of A020669 (which allows for a=0 and/or b=0). See there for further references. See A155560 ff for intersection of sequences of type (a^2 + k b^2).

Also, subsequence of A000408 (with 5b^2 = b^2 + (2b)^2).

Links

Table of n, a(n) for n=1..58.

Example

a(1) = 6 = 1^2 + 5*1^2 is the least number that can be written as A+5B where A,B are positive squares.
a(2) = 9 = 2^2 + 5*1^2 is the second smallest number that can be written in this way.

Mathematica

formQ[n_] := Reduce[a > 0 && b > 0 && n == a^2 + 5 b^2, {a, b}, Integers] =!= False; Select[ Range[300], formQ] (* Jean-François Alcover, Sep 20 2011 *)
Timing[mx = 300; limx = Sqrt[mx]; limy = Sqrt[mx/5]; Select[Union[Flatten[Table[x^2 + 5 y^2, {x, limx}, {y, limy}]]], # <= mx &]] (* T. D. Noe, Sep 20 2011 *)

Prog

(PARI) isA154778(n, /* use optional 2nd arg to get other analogous sequences */c=5) = { for( b=1, sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1, 300, isA154778(n) & print1(n", "))

Crossrefs

Cf. A033205 (subsequence of primes). [From R. J. Mathar, Jan 26 2009]

Sequence in context: A316011 A316012 A316013 * A316014 A106350 A217851

Adjacent sequences: A154775 A154776 A154777 * A154779 A154780 A154781

Keyword

easy,nonn

Author

M. F. Hasler, Jan 24 2009

Status

approved