A145236 - OEIS

%I #10 Mar 04 2019 10:33:04

%S 2,1,1,3,5,5,9,9,13,17,19,23,25,27,31,35,41,41,47,51,51,57,61,65,73,

%T 75,77,81,83,85,99,101,107,109,117,119,125,129,133,139,145,145,155,

%U 157,161,163,173,183,187,189,193,199,201,209,215,221,225,227,233,237,239,247

%N a(n) is the least positive integer such that if p_n is the n-th prime then (ceiling(sqrt(a(n)*p_n)))^2 - a(n)*p_n is a perfect square.

%C Conjectures: 1) for n >= 2, the sequence does not decrease; 2) for n > 1, a(n) is odd; 3) a(n) can be equal to a(n+1) only for twins: p_(n+1) - p_n = 2 (although there also exist twins for which a(n) < a(n+1)).

%C All these conjectures are proved using the formula a(n) = p_n - 2*floor(sqrt(2p_n)) + 2, n > 1. See also A145701 and A145714. - _Vladimir Shevelev_, Oct 18 2008

%p A145236 := proc(n) local p,k,a ; p := ithprime(n) ; for k from 1 do ceil(sqrt(ceil(k*p))) ; a := %^2-k*p ; if issqr(a) then return k ; end if; end do: end proc:

%p for n from 1 do printf("%d,\n",A145236(n)) ; end do: # _R. J. Mathar_, Aug 02 2010

%Y Cf. A145016, A145022, A145023, A145047, A145048, A145049, A145050, A145215.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Oct 05 2008, Oct 07 2008

%E a(12)=23 (not 21). - _Vladimir Shevelev_, Oct 16 2008

%E Extended by _R. J. Mathar_, Aug 02 2010