A155715 - OEIS

%I #6 Jul 14 2012 11:32:23

%S 2,17,73,73,241,241,1009,1009,1009,1009,7561,7561,21961,32356,32356,

%T 32356,44641,44641,349924,349924,349924,349924,1399696,1399696,

%U 1399696,3027249,3027249,3027249,4349601,4349601,18567396,18567396,18567396

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

%C 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.

%C a(n) > 10^9 for n >= 47. [From _Donovan Johnson_, Sep 29 2009]

%H Donovan Johnson, <a href="/A155715/b155715.txt">Table of n, a(n) for n=1..46</a>

%e 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).

%o (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--)

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

%K nonn

%O 1,1

%A _M. F. Hasler_, Jan 27 2009

%E a(23)-a(46) and b-file from _Donovan Johnson_, Sep 29 2009