A204065 -id:A204065 - cp's OEIS frontend

A069003 Smallest integer d such that n^2 + d^2 is a prime number.

1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 4, 7, 2, 1, 2, 1, 2, 5, 6, 1, 4, 5, 8, 1, 4, 1, 2, 5, 4, 11, 4, 3, 2, 5, 2, 1, 2, 3, 10, 1, 4, 5, 8, 9, 2, 5, 2, 13, 4, 7, 4, 3, 10, 1, 4, 1, 2, 3, 6, 13, 10, 3, 32, 9, 2, 1, 2, 5, 10, 3, 6, 5, 2, 1, 4, 5, 10, 7, 4, 7, 4, 3, 18, 1, 2, 9, 2, 3, 4, 1, 4, 7, 8, 1, 2, 5, 2, 3, 4, 3

Offset: 1

T. D. Noe, Apr 02 2002

With i being the imaginary unit, n + di is the smallest Gaussian prime with real part n and a positive imaginary part. Likewise for n - di. See A002145 for Gaussian primes with imaginary part 0. - Alonso del Arte, Feb 07 2011
Conjecture: a(n) does not exceed 4*sqrt(n+1) for any positive integer n. - Zhi-Wei Sun, Apr 15 2013
Conjecture holds for the first 15*10^6 terms. - Joerg Arndt, Aug 19 2014
Infinitely many d exist such that n^2 + d^2 is prime, under Schinzel's Hypothesis H; see Sierpinski (1988), p. 221. - Jonathan Sondow, Nov 09 2015

			a(5)=2 because 2 is the smallest integer d such that 5^2+d^2 is a prime number.

W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Gaussian Prime.
Wikipedia, Schinzel's Hypothesis H.

Cf. A068486 (lists the prime numbers n^2 + d^2).
Cf. A185636, A204065.
Cf. A239388, A239389 (record values).
Cf. A053000.

f:= proc(n) local d;
     for d from 1+(n mod 2) by 2 do
       if isprime(n^2+d^2) then return d fi
     od
end proc:
f(1):= 1:
map(f, [$1..1000]); # Robert Israel, Jul 06 2015

imP4P[n_] := Module[{k = 1}, While[Not[PrimeQ[n^2 + k^2]], k++]; k]; Table[imP4P[n], {n, 50}] (* Alonso del Arte, Feb 07 2011 *)

a(n)=my(k);while(!isprime(n^2+k++^2),);k \\ Charles R Greathouse IV, Mar 20 2013

A227898 Number of primes p < n with p + 6 and n + (n - p)^2 both prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 4, 3, 2, 2, 3, 3, 3, 3, 3, 4, 4, 2, 2, 3, 3, 3, 2, 2, 5, 5, 2, 5, 4, 2, 4, 5, 2, 7, 5, 3, 4, 5, 3, 3, 4, 4, 3, 5, 4, 9, 9, 2, 5, 3, 4, 8, 6, 2, 5, 8, 3, 4, 7, 3, 10, 5, 2, 7, 4, 5, 10, 6, 4, 6, 6, 2, 6, 8, 3, 6, 5, 3, 6, 6, 5, 9, 4, 5, 7, 5, 4, 9, 10

Offset: 1

Zhi-Wei Sun, Oct 14 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) For any integer n > 5, there is a prime p with p + 6 and n*(n - p) - 1 both prime.

			a(6) = 1 since 5, 5 + 6 = 11 and 6 + (6 - 5)^2 = 7 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A023201, A185636, A204065, A227899, A230261.

a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[n+(n-Prime[i])^2],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 2, 1, 2, 3, 2, 3, 3, 2, 3, 2, 3, 1, 1, 3, 2, 4, 2, 3, 3, 3, 4, 1, 2, 6, 2, 4, 2, 3, 5, 4, 2, 3, 4, 4, 4, 4, 2, 1, 2, 4, 2, 4, 2, 6, 7, 5, 3, 3, 9, 2, 3, 3, 2, 4, 4, 3, 1, 2, 8, 3, 6, 2, 2, 8, 4, 7, 2, 2, 5, 2, 3, 3, 2, 8, 3, 3, 1, 4, 7, 5, 9, 2, 2, 5

Offset: 1

Zhi-Wei Sun, Oct 14 2013

nonn

Conjecture: a(n) > 0 for all n > 3.

			a(5) = 1 since 5 = 3 + 2, and the three numbers 3, 3*3 - 4 = 5 and 5^2 + (5-3)^2 = 29 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A069003, A185636, A204065, A227898, A230223.

a[n_]:=Sum[If[PrimeQ[3Prime[i]-4]&&PrimeQ[n^2+(n-Prime[i])^2],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A261437 Least positive integer k such that kn+1 = prime(p) and k^2n+1 = prime(q) for some pair of primes p and q.

Original entry on oeis.org

2, 1, 286, 1, 7290, 21, 18, 2472, 12, 1, 20460, 20, 20692, 105, 4392, 1, 96816, 1327, 360, 264, 19850, 2734, 1854, 5293, 930, 29526, 98, 622, 9222, 1, 6816, 924, 61614, 70, 53760, 45, 32190, 9687, 5510, 1, 128070, 148, 8772, 23478, 404, 801, 1830, 5, 9912, 7662, 1100, 8211, 1116, 9997, 630, 4965, 936, 1, 87570, 759

Offset: 1

Zhi-Wei Sun, Aug 18 2015

nonn

Conjecture: (i) If n > 0 and r are relatively prime integers, then there are infinitely many positive integers k such that k*n+r = prime(p) for some prime p.
(ii) Let r be 1 or -1. For any integer n > 0, there is a positive integer k such that k*n+r = prime(p) and k^2*n+1 = prime(q) for some primes p and q.
(iii) For any integer n > 0, there is a positive integer k such that n+k = prime(p) and n+k^2 = prime(q) for some primes p and q.
Note that part (i) is a refinement of Dirichlet's theorem on primes in arithmetic progressions. Also, part (ii) implies that a(n) exists for any n > 0.

			a(3) = 286 since 286*3+1 = 859 = prime(149) with 149 prime, and 286^2*3+1 = 245389 = prime(21661) with 21661 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000290, A034693, A053989, A185636, A204065.

PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0;Label[bb];k=k+1;If[PQ[k*n+1]&&PQ[k^2*n+1],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]

A224030 a(n) = |{0

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 15 2013

nonn

Conjecture: a(n)>0 for all n>4.
This has been verified for n up to 10^8.
We also conjecture that for any integer n>1 there is an integer 0

			a(7) = 1 since 2*7+5 = 19 and 2*7^3+5^3 = 811 are both prime.
a(57) = 1 since 2*57+23 = 137 and 2*57^3+23^3 = 382553 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A185636, A204065, A220413.

a[n_]:=a[n]=Sum[If[PrimeQ[2n+k]==True&&PrimeQ[2n^3+k^3]==True,1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

Original entry on oeis.org

5, 3, 3, 5, 3, 5, 3, 7, 11, 5, 3, 5, 3, 7, 17, 5, 3, 5, 3, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 13, 11, 7, 19, 5, 3, 7, 17, 5, 3, 5, 3, 7, 17, 5, 3, 23, 11, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 31, 11, 7, 19, 5, 3, 7, 11, 5, 3, 5, 3, 13, 17, 7, 19, 5, 3, 7, 17, 5, 3, 23, 17, 7, 11, 5, 3, 29, 11, 13, 11, 7, 19, 5, 3, 7, 11, 5

Offset: 1

Zhi-Wei Sun, Nov 18 2013

nonn

Conjecture: a(n) > 0 for all n > 0. Moreover, for any integer n > 1 there exists a prime p < 2*sqrt(n)*log(7n) such that n + (p-1)*(p-3)/8 is prime.

This implies that any integer n > 1 can be written as (p-1)/2 + q with q a positive integer, and p and (p^2-1)/8 + q both prime.

			a(1) = 5 since neither 1 + (2-1)*(2-3)/8 = 7/8 nor 1 + (3-1)*(3-3)/8 = 1  is prime, but 1 + (5-1)*(5-3)/8 = 2 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A185636, A204065, A227898, A228425, A231201, A231557.

Do[Do[If[PrimeQ[n+(Prime[k]-1)(Prime[k]-3)/8],Goto[aa]],{k,1,PrimePi[n+4]}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

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cp's OEIS Frontend

A069003 Smallest integer d such that n^2 + d^2 is a prime number.

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A227898 Number of primes p < n with p + 6 and n + (n - p)^2 both prime.

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A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

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A261437 Least positive integer k such that k*n+1 = prime(p) and k^2*n+1 = prime(q) for some pair of primes p and q.

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A224030 a(n) = |{0

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A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

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A261437 Least positive integer k such that kn+1 = prime(p) and k^2n+1 = prime(q) for some pair of primes p and q.