A155715 Least number expressible as a^2 + k b^2 with positive integers a,b, for each k=1,...,n.

2, 17, 73, 73, 241, 241, 1009, 1009, 1009, 1009, 7561, 7561, 21961, 32356, 32356, 32356, 44641, 44641, 349924, 349924, 349924, 349924, 1399696, 1399696, 1399696, 3027249, 3027249, 3027249, 4349601, 4349601, 18567396, 18567396, 18567396

Offset: 1

Author

M. F. Hasler, Jan 27 2009

Keywords

• nonn

Comments

Sequence A028372 considers primes with this property, but allowing for nonzero a,b (which obviously is irrelevant for n>2). Up to n=13, the terms of the present sequence are prime without imposing it explicitely and thus coincide with A028372 except for n=2.

a(n) > 10^9 for n >= 47. [From Donovan Johnson, Sep 29 2009]

Examples

			a(1) = 2 = 1^2 + 1^2 is the least number of the sequence A000404 (sum of positive squares). a(2) = 17 = 1^2 + 4^2 = 3^2 + 2*2^2 is the least number in sequence A000404 to be in sequence A154777 (a^2+2b^2)as well. a(3) = 73 = 3^2 + 8^2 = 1^2 + 2*6^2 = 5^2 + 3*4^2 is the least number in the intersection of sequences A000404, A154777 and A092572 (a^2+3b^2).
		

Links

• Donovan Johnson, Table of n, a(n) for n=1..46

Crossrefs

Cf. A028372, A000404, A154777, A092572, A097268, A154778, A155707-A155716, A155560-A155578.

Programs

PARI

k=1; for( n=1,10^9, forstep( c=k,1,-1, for( b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & next(2));next(2)); print1(n",");k++;n--)

Extensions

a(23)-a(46) and b-file from Donovan Johnson, Sep 29 2009