A070317 -id:A070317 - cp's OEIS frontend

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

Offset: 0

N. J. A. Sloane, Feb 21 2000

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

J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
R. K. Guy, Unsolved Problems in Number Theory, Section A1.

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

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

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

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

nxt[n_]:=Module[{n2=n^2},NextPrime[n2]-n2]
nxt/@Range[0,100]  (* Harvey P. Dale, Dec 20 2010 *)

A053000(n)=nextprime(n^2)-n^2  \\ M. F. Hasler, Mar 23 2013

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

a(n) = A013632(n^2). - Robert Israel, Jul 06 2015

More terms from James Sellers, Feb 22 2000

A058055 a(n) is the smallest positive number m such that m^2 + n is the next prime > m^2.

Original entry on oeis.org

1, 3, 8, 5, 12, 11, 18, 51, 82, 49, 234, 23, 42, 75, 86, 231, 174, 107, 288, 63, 80, 69, 102, 325, 166, 765, 128, 143, 822, 727, 276, 597, 226, 835, 702, 461, 254, 693, 592, 797, 1284, 349, 370, 2337, 596, 645, 3012, 1033, 590, 4083, 1490, 757, 882, 833, 1668

Offset: 1

Labos Elemer, Nov 20 2000

nonn

The primes are in A058056.

			n=6: a(6)=11 and 11^2+6 is 127, a prime; n=97: a(97) = 2144 and 2144^2+97 = 4596833, the least prime of the form m^2+97.

Zak Seidov, Table of n, a(n) for n = 1..500 (first 400 terms from T. D. Noe)

Cf. A053000, A070316, A070317.

See A085099, A215249 for other versions.

for m from 1 to 10^5 do
   r:= nextprime(m^2)-m^2;
   if not assigned(R[r]) then R[r]:= m end if;
end do:
J:= map(op,{indices(R)}):
N:= min({$1..J[-1]} minus J)-1:
[seq(R[j],j=1..N)]; # Robert Israel, Aug 10 2012

nn = 100; t = Table[0, {nn}]; found = 0; m = 0; While[found < nn, m++; k = NextPrime[m^2] - m^2; If[k <= nn && t[[k]] == 0, t[[k]] = m; found++]]; t (* T. D. Noe, Aug 10 2012 *)

R = {}   # After Robert Israel's Maple script.
for m in (1..2^12) :
    r = next_prime(m^2) - m^2
    if r not in R : R[r] = m
L = sorted(R.keys())
for i in (1..len(L)-1) :
    if L[i] != L[i-1]+1 : break
[R[k] for k in (1..i)]  # Peter Luschny, Aug 11 2012

a(n) = Min{ m > 0 | m^2 + n is the next prime after m^2}.

A053000(a(n)) = n. - Zak Seidov, Apr 12 2013

Definition corrected by Zak Seidov, Mar 03 2008, and again by Franklin T. Adams-Watters, Aug 10 2012

A070316 Numbers n such that nextprime(n^2)-n^2 (n>=1) sets a new record.

Original entry on oeis.org

1, 3, 5, 11, 18, 23, 42, 63, 69, 102, 128, 143, 226, 254, 349, 370, 590, 757, 833, 873, 1110, 1165, 5018, 14908, 49495, 87598, 106359, 185179, 269697, 558269, 1070246, 1673629, 4292936, 43957056, 148793437, 982920306, 1569443693, 4556758439

Offset: 1

Donald S. McDonald, May 11 2002

nonn

			nextprime(63^2) - 63^2 = 3989 - 3969 = 20, giving the terms 63 in the present sequence and 20 in A070317.

Charles R Greathouse IV, Table of n, a(n) for n = 1..45, Dec 29 2007

Positions of record values in A053000 for n >= 1. Cf. A070317.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; d = 0; Do[m = n; a = NextPrim[n^2] - n^2; If[a > d, d = a; Print[n]], {n, 1, 10^8}]

A070316(n,show_all=0)={my(k=0,r=0);for(n=1,n,until(nextprime(k++^2)-k^2>r ,); r=nextprime(k^2)-k^2; show_all&print1(k","));k} \\ M. F. Hasler, Mar 23 2013

Edited by N. J. A. Sloane and Robert G. Wilson v, May 11 2002
More terms from Ralf Stephan, Oct 14 2002
Further terms from Charles R Greathouse IV, Jun 16 2007
Typo in a(38) corrected by M. F. Hasler, Mar 23 2013

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A053000 a(n) = (smallest prime > n^2) - n^2.

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A058055 a(n) is the smallest positive number m such that m^2 + n is the next prime > m^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A070316 Numbers n such that nextprime(n^2)-n^2 (n>=1) sets a new record.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions