%I #52 Nov 21 2023 19:31:15

%S 2,1,1,2,1,4,1,4,3,2,1,6,5,4,1,2,1,4,7,6,1,2,3,12,1,6,1,4,3,12,7,6,7,

%T 2,7,4,1,4,3,2,1,12,13,12,13,2,13,4,5,10,3,8,3,10,1,12,1,2,7,10,7,6,3,

%U 20,3,4,1,4,13,22,3,10,5,4,1,14,3,10,5,6,21,2,9,10,1,4,15,4,9,6,1,6,3,14

%N a(n) = (smallest prime > n^2) - n^2.

%C Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2.

%C Record values are listed in A070317, their indices in A070316. - _M. F. Hasler_, Mar 23 2013

%C Conjecture: a(n) <= 1+phi(n) = 1+A000010(n), for n>0. This improves on Oppermann's conjecture, which says a(n) < n. - _Jianglin Luo_, Sep 22 2023

%D J. R. Goldman, The Queen of Mathematics, 1998, p. 82.

%H T. D. Noe, <a href="/A053000/b053000.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A013632(n^2). - _Robert Israel_, Jul 06 2015

%p A053000 := n->nextprime(n^2)-n^2;

%t nxt[n_]:=Module[{n2=n^2},NextPrime[n2]-n2]

%t nxt/@Range[0,100] (* _Harvey P. Dale_, Dec 20 2010 *)

%o (PARI) A053000(n)=nextprime(n^2)-n^2 \\ _M. F. Hasler_, Mar 23 2013

%o (Magma) [NextPrime(n^2) - n^2: n in [0..100]]; // _Vincenzo Librandi_, Jul 06 2015

%o (Python)

%o from sympy import nextprime

%o def a(n): nn = n*n; return nextprime(nn) - nn

%o print([a(n) for n in range(94)]) # _Michael S. Branicky_, Feb 17 2022

%Y Cf. A007491, A013632, A053001, A014085, A070316, A085099, A058055, A069003.

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_, Feb 21 2000

%E More terms from _James A. Sellers_, Feb 22 2000