A056896 -id:A056896 - cp's OEIS frontend

A056898 a(n) = smallest number m such that m^2+n is prime.

1, 0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 4, 1, 0, 7, 2, 9, 2, 1, 0, 1, 0, 3, 2, 3, 6, 1, 0, 3, 2, 1, 0, 1, 0, 3, 4, 1, 0, 5, 2, 3, 4, 1, 0, 5, 2, 9, 2, 1, 0, 1, 0, 3, 2, 3, 6, 1, 0, 9, 2, 1, 0, 1, 0, 3, 2, 5, 6, 1, 0, 3, 4, 1, 0, 5, 2, 9, 4, 1, 0, 7, 4, 3, 2, 3, 6, 1, 0, 3, 2

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(8) = 3 since 3^2+8 = 17 which is prime.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000040, A002496, A056892, A056893, A056894, A056895, A056896, A056897.

A056898(n) = { my(m=0); while(!isprime((m*m)+n),m++); (m); }; \\ Antti Karttunen, Mar 04 2018

a(n) = sqrt(A056896(n)-n) = sqrt(A056897(n)).

For p a prime: a(p) = 0 (and a(p-1) = 1 if p<>3).

A056897 Smallest square where a(n)+n is prime.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 9, 4, 1, 0, 1, 0, 9, 4, 1, 0, 1, 0, 9, 16, 1, 0, 49, 4, 81, 4, 1, 0, 1, 0, 9, 4, 9, 36, 1, 0, 9, 4, 1, 0, 1, 0, 9, 16, 1, 0, 25, 4, 9, 16, 1, 0, 25, 4, 81, 4, 1, 0, 1, 0, 9, 4, 9, 36, 1, 0, 81, 4, 1, 0, 1, 0, 9, 4, 25, 36, 1, 0, 9, 16, 1, 0, 25, 4, 81, 16, 1, 0, 49, 16, 9, 4, 9

Offset: 0

Henry Bottomley, Jul 05 2000

nonn

			a(8)=9 since 9 is a square and 9+8=7 which is a prime

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000040, A002496, A056892-A056898.

With[{sqs=Range[0,20]^2},Table[SelectFirst[sqs,PrimeQ[n+#]&],{n,100}]] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, May 07 2016 *)

a(n) =A056896(n)-n =A056898(n)^2

A056895 If the smallest prime with a square excess of n is p then a(n)^2 = p - n.

Original entry on oeis.org

1, 1, 2, 3, 6, 5, 4, 9, 8, 7, 6, 7, 10, 15, 8, 9, 24, 11, 12, 21, 16, 13, 12, 13, 16, 15, 14, 17, 18, 31, 20, 27, 20, 23, 18, 19, 22, 21, 20, 23, 24, 23, 24, 27, 28, 29, 30, 25, 38, 39, 26, 31, 30, 35, 28, 45, 34, 31, 42, 31, 34, 33, 32, 33, 36, 35, 34, 75, 40, 37, 36, 41, 48, 45

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(4)=3 because the smallest prime with a square excess of 4 is 13 and 13 - 4 = 3^2.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000040, A000196, A002496, A048760, A053186.

Cf. A056892, A056893, A056894, A056896, A056897, A056898.

a = {}; Do[p = 2; While[n != p - (r = Floor@Sqrt[p])^2, p = NextPrime[p]]; AppendTo[a, r], {n, 74}]; a (* Ivan Neretin, May 02 2019 *)

a(n) = {my(p=2); while(n != p-sqrtint(p)^2, p = nextprime(p+1)); sqrtint(p - n);} \\ Michel Marcus, May 05 2019

a(n) = sqrt(A056893(n)-n) = A000196(A056893(n)) = sqrt(A056894(n)).

A059843 a(n) is the smallest prime p such that p-n is a nonzero square.

Original entry on oeis.org

2, 3, 7, 5, 41, 7, 11, 17, 13, 11, 47, 13, 17, 23, 19, 17, 53, 19, 23, 29, 37, 23, 59, 73, 29, 107, 31, 29, 173, 31, 47, 41, 37, 43, 71, 37, 41, 47, 43, 41, 617, 43, 47, 53, 61, 47, 83, 73, 53, 59, 67, 53, 89, 79, 59, 137, 61, 59, 383, 61, 97, 71, 67, 73, 101, 67, 71, 149, 73

Offset: 1

Labos Elemer, Feb 26 2001

nonn

			For n = 17, let P = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,...} be the set of primes, then P - 17 = {-15,...,-4,0,2,6,12,14,20,24,26,30,36,...}. The first positive square in P - 17 is 36 with p = 53, so a(17) = 53. The square arising here is usually 1.

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

These terms arise in A002496, A056899, A049423, A005473, A056905, A056909 as first or 2nd entries depending on offset.
Cf. A002496, A056899, A049423, A005473, A056905, A056909.
Cf. A056896 (where p-n can be 0).

SearchLimit := 100;
for n from 1 to 400 do
k := 0: c := true:
while(c and k < SearchLimit) do
    k := k + 1:
    c := not isprime(k^2+n):
end do:
if k = SearchLimit then error("Search limit reached!") fi;
a[n] := k^2 + n end do: seq(a[j], j=1..400);
# Edited and SearchLimit introduced by Peter Luschny, Feb 05 2019

spsq[n_]:=Module[{p=NextPrime[n]},While[!IntegerQ[Sqrt[p-n]],p= NextPrime[ p]];p]; Array[spsq,70] (* Harvey P. Dale, Nov 10 2017 *)

for(n=1, 100, for(k=1, 100, if(isprime(k^2+n), print1(k^2+n, ", "); break()))) \\ Jianing Song, Feb 04 2019

a(n) = forprime(p=n,, if ((p-n) && issquare(p-n), return (p))); \\ Michel Marcus, Feb 05 2019

a(n) = min{p : p - n = x^2 for some x > 0, p is prime}.

Does a(n) exist for all n? - Jianing Song, Feb 04 2019

cp's OEIS Frontend

A056898 a(n) = smallest number m such that m^2+n is prime.

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A056897 Smallest square where a(n)+n is prime.

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A056895 If the smallest prime with a square excess of n is p then a(n)^2 = p - n.

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A059843 a(n) is the smallest prime p such that p-n is a nonzero square.

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