A058055 -id:A058055 - 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

A085099 Least natural number k such that k^2 + n is prime.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 4, 1, 6, 7, 2, 9, 2, 1, 12, 1, 4, 3, 2, 3, 6, 1, 2, 3, 2, 1, 24, 1, 2, 3, 4, 1, 6, 5, 2, 3, 4, 1, 6, 5, 2, 9, 2, 1, 18, 1, 6, 3, 2, 3, 6, 1, 2, 9, 2, 1, 6, 1, 4, 3, 2, 5, 6, 1, 2, 3, 4, 1, 12, 5, 2

Offset: 1

Jason Earls, Aug 10 2003

First values of k and n such that k > 100 are: k=114, n=6041; for k > 200: k=210, n=26171; for k > 300: k=357, n=218084; for k > 400: k=402, n=576239.
Additionally, for k > 500: k=585, n=3569114; for k > 600: k=630, n=3802301; for k > 700: k=744, n=24307841; for k > 800: k=855, n=25051934; for k > 900: k=1008, n=54168539. Other cases k > 900: k=945, n=74380946, k=915, n=89992964, k=939, n=118991066. - Zak Seidov, May 23 2007
It is easily proved that for n > 2, a(n) >= A089128(n+1). The first inequality is a(21) = 4. - Franklin T. Adams-Watters, May 16 2018

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A059843, A200926, A058055, A053000, A089128.

a:= proc(n) local d, t; d, t:= 1, n+1; while not
      isprime(t) do d:= d+2; t:= t+d od; (d+1)/2
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 04 2019

Table[i = 1; While[! PrimeQ[i^2 + n], i++]; i, {n, 85}] (* Jayanta Basu, Apr 24 2013 *)

a(n)=my(k); while(!isprime(k++^2+n),); k \\ Charles R Greathouse IV, Jul 17 2016

a(n) = sqrt(A059843(n) - n). - Zak Seidov, Nov 24 2011

A058056 a(n) = p is the smallest prime such that p = n + h(n)^2 and p is the first prime following h(n)^2. The smallest immediate post-square primes with distance n = p - h(n)^2.

Original entry on oeis.org

2, 11, 67, 29, 149, 127, 331, 2609, 6733, 2411, 54767, 541, 1777, 5639, 7411, 53377, 30293, 11467, 82963, 3989, 6421, 4783, 10427, 105649, 27581, 585251, 16411, 20477, 675713, 528559, 76207, 356441, 51109, 697259, 492839, 212557, 64553, 480287, 350503, 635249

Offset: 1

Labos Elemer, Nov 20 2000

nonn

The primes generated by the numbers in A058055.

			For n=5, a(5) = 149 = 5+144 = 5+12^2; although 41 = 5+36 = 5+k^2 but between 41 and 36 further prime occurs 37 while no more primes are between 144 and 149. n=7 a(7) = 331 = 324+7 = 18*18+7 and 331 = nextprime(324); numerous smaller primes (like {7, 11, 23, 43, 71, 107, 151, 263} = 7 + {0, 4, 16, 36, 64, 100, 144, 256}) have q = 7+k^2 form so that q is not the nextprime(7+k^2), 324 is the smallest square of this kind.

T. D. Noe, Table of n, a(n) for n = 1..400

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^2 + k; found++]]; t (* T. D. Noe, Aug 12 2012 *)

A215249 a(n) is the smallest number m >= n such that m^2 + n is a prime.

Original entry on oeis.org

1, 3, 4, 5, 6, 11, 8, 9, 10, 11, 24, 13, 16, 15, 16, 21, 24, 19, 20, 21, 40, 25, 24, 35, 26, 69, 28, 33, 30, 31, 34, 39, 56, 35, 48, 49, 38, 39, 50, 41, 54, 47, 44, 45, 46, 49, 48, 53, 50, 57, 56, 53, 54, 55, 56, 69, 64, 59, 60, 67, 64, 105, 64, 65, 66, 71, 68, 75

Offset: 1

N. J. A. Sloane, Aug 10 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A058055, A085099.

Table[m = n; While[! PrimeQ[m^2 + n], m++]; m, {n, 100}] (* T. D. Noe, Aug 12 2012 *)

cp's OEIS Frontend

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

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A085099 Least natural number k such that k^2 + n is prime.

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A058056 a(n) = p is the smallest prime such that p = n + h(n)^2 and p is the first prime following h(n)^2. The smallest immediate post-square primes with distance n = p - h(n)^2.

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A215249 a(n) is the smallest number m >= n such that m^2 + n is a prime.

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