A056896 -id:A056896

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A056898

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

LINKS

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

FORMULA

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

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

EXAMPLE

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

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2000

STATUS

approved

A056897

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A000040, A002496, A056892-A056898.

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2000

STATUS

approved

A056895

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2000

STATUS

approved

A059843

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

Labos Elemer, Feb 26 2001

STATUS

approved

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