A053000 - OEIS

A053000

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

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, 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, 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

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OFFSET

0,1

COMMENTS

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

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

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

REFERENCES

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

R. K. Guy, Unsolved Problems in Number Theory, Section A1.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

FORMULA

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

MAPLE

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

MATHEMATICA

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

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

PROG

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

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

(Python)

from sympy import nextprime

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

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

CROSSREFS

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

Sequence in context: A276376 A165585 A082506 * A002070 A326376 A106052

Adjacent sequences: A052997 A052998 A052999 * A053001 A053002 A053003

KEYWORD

nonn,easy,nice

AUTHOR

_N. J. A. Sloane_, Feb 21 2000

EXTENSIONS

More terms from _James A. Sellers_, Feb 22 2000

STATUS

approved