A185636 -id:A185636 - 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

A238457 a(n) = |{0 < k <= n: p(n) + k is prime}|, where p(.) is the partition function (A000041).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 27 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 3, 4, 5, 30, 109. Also, for each n > 2 there is a positive integer k <= n+3 such that p(n) - k is prime.

(ii) For the strict partition function q(.) given by A000009, we have |{0 < k <= n: q(n) + k is prime}| > 0 for all n > 0 and |{0 < k <= n: q(n) - k is prime}| > 0 for all n > 4.

			a(5) = 1 since p(5) + 4 = 7 + 4 = 11 is prime.
a(30) = 1 since p(30) + 19 = 5604 + 19 = 5623 is prime.
a(109) = 1 since p(109) + 63 = 541946240 + 63 = 541946303 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000009, A000040, A000041, A185636.

p[n_,k_]:=PrimeQ[PartitionsP[n]+k]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]
Table[a[n],{n,1,100}]

A204065 Least nonnegative integer k with n+k and n+k^2 both prime.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 3, 16, 1, 0, 7, 4, 15, 2, 1, 0, 1, 0, 9, 8, 3, 6, 1, 0, 3, 2, 1, 0, 1, 0, 3, 8, 1, 0, 5, 10, 3, 10, 1, 0, 5, 4, 15, 2, 1, 0, 1, 0, 21, 4, 3, 6, 1, 0, 15, 2, 1, 0, 1, 0, 33, 8, 25, 6, 1, 0, 3, 16, 1, 0, 5, 4, 15, 14, 1, 0, 7, 6, 9, 4, 3, 6, 1, 0, 3, 2, 1

Offset: 1

Zhi-Wei Sun, Jan 09 2013

nonn

Conjecture: For any n > 0 not among 1, 21, 326, 341, 626, we have a(n) < sqrt(n)*log(n). If n > 626 is not equal to 971, then n+k and n+k^2 are both prime for some 0< k < sqrt(n)*log(n). Also, n+k^2 is prime for some 0< k <= sqrt(n) if n > 43181.

Obviously, a(n)=0 iff n is a prime. - M. F. Hasler, Jan 11 2013

			a(8)=3 since 8+3 and 8+3^2 are both prime, but none of 8, 8+1, 8+2 is prime.

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

Cf. A185636, A071558.

Do[Do[If[PrimeQ[n+k]==True&&PrimeQ[n+k^2]==True,Print[n," ",k];Goto[aa]],{k,0,n}];
Label[aa];Continue,{n,1,100}]

a(n)=my(k=0); while(!isprime(n+k) || !isprime(n+k^2), k++); k \\ - M. F. Hasler, Jan 11 2013

A210531 Number of nonnegative integers k

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 2, 2, 2, 3, 2, 2, 4, 5, 4, 2, 3, 7, 5, 1, 2, 7, 4, 2, 7, 5, 6, 1, 5, 9, 4, 4, 6, 9, 9, 2, 5, 12, 9, 3, 5, 6, 8, 5, 6, 13, 4, 2, 8, 6, 11, 6, 11, 14, 8, 2, 4, 7, 4, 5, 7, 29, 8, 3, 5, 8, 11, 4, 13, 16, 13, 2, 7, 12, 13, 6, 10, 16, 10, 6, 15, 9, 13, 3, 9, 20, 11, 8, 11, 20, 9, 2, 8, 22, 14, 6, 15, 15

Offset: 1

Zhi-Wei Sun, Jan 28 2013

nonn

Conjecture: a(n)>0 for all n>0. Moreover, if n>0 is different from 74, 138, 166, 542, then n+k^3 is practical for some 0<=k<=sqrt(n)*log(n); if n is not equal to 102, then n+k and n+k^3 are both practical for some k=0,...,n-1.

Zhi-Wei Sun also conjectured that any integer n>1 can be written as x^3+y (x,y>0) with 2x and 4xy both practical.

			a(22)=1 since 22+2^3=30 is practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A005153, A185636, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[n+k^3]==True,1,0],{k,0,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A185150 Number of odd primes p between n^2 and (n+1)^2 with (n/p) = 1, where (-) is the Legendre symbol.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 2, 3, 3, 1, 4, 2, 4, 3, 5, 7, 2, 3, 4, 6, 5, 3, 3, 4, 8, 5, 4, 5, 4, 4, 6, 6, 6, 4, 9, 9, 7, 7, 5, 6, 7, 5, 9, 5, 7, 3, 9, 6, 10, 6, 10, 6, 8, 8, 7, 7, 10, 3, 12, 8, 7, 10, 8, 14, 11, 7, 10, 10, 5, 9, 11, 8, 7, 9, 9, 18, 11, 11, 12, 9, 20, 6, 13, 6, 10, 9, 13, 9, 8, 10, 10, 12, 12, 6, 13, 9, 12, 12, 8, 23

Offset: 1

Zhi-Wei Sun, Dec 29 2012

nonn

Conjecture: a(n)>0 for all n>0.
This is a refinement of Legendre's conjecture that for each n=1,2,3,... the interval (n^2,(n+1)^2) contains a prime.
We have verified the conjecture for n up to 10^9.
Zhi-Wei Sun also made some similar conjectures involving primes and Legendre symbols, below are few examples:
(1) If n>10 then there is a prime p between n^2 and (n+1)^2 with (n/p) = ((1-n)/p) = 1. If n>2 is different from 7 and 17, then there is a prime p between n^2 and (n+1)^2 with (n/p) = ((n+1)/p) = 1. If n>1 is not equal to 27, then there is a prime p between n^2 and (n+1)^2 with (n/p) = ((n+2)/p) = 1.
(2) If n>2 is different from 6, 12, 58, then there is a prime p between n^2 and n^2+n such that (n/p) = 1. If n>20 is not a square, and different from 37 and 77, then there is a prime p between n^2 and n^2+n such that (n/p) = -1.
(3) For each n=15,16,... there is a prime p between n and 2n such that (n/p) = 1. If n>0 is not a square, then
there is a prime p between n and 2n such that (n/p) = -1.

			a(10)=1 since 107 is the only prime p between 10^2 and 11^2 with (10/p) = 1.

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

Cf. A014085, A185636.

a[n_]:=a[n]=Sum[If[n^2+k>2&&PrimeQ[n^2+k]==True&&JacobiSymbol[n,n^2+k]==1,1,0],{k,1,2n}]
Do[Print[n," ",a[n]],{n,1,100}]

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}]

A237582 a(n) = |{0 < k < n: pi(n + k^2) is prime}|, where pi(.) is given by A000720.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 3, 2, 3, 4, 1, 2, 2, 3, 6, 6, 5, 5, 5, 5, 6, 7, 7, 6, 5, 6, 5, 6, 7, 8, 9, 8, 10, 9, 8, 6, 6, 6, 6, 7, 9, 9, 10, 11, 11, 13, 11, 9, 9, 10, 10, 8, 6, 6, 5, 4, 8, 9, 10, 12, 11, 14, 15, 15, 15, 12, 14, 15, 17, 16, 13, 11, 11, 13, 16, 18, 24, 25, 20

Offset: 1

Zhi-Wei Sun, Feb 09 2014

nonn

Conjecture: (i) For each a = 2, 3, ... there is a positive integer N(a) such that for any integer n > N(a) there is a positive integer k < n with pi(n + k^a) prime. In particular, we may take (N(2), N(3), ..., N(9)) = (1, 1, 9, 26, 8, 9, 18, 1).
(ii) If n > 6, then pi(n^2 + k^2) is prime for some 0 < k < n. If n > 27, then pi(n^3 + k^3) is prime for some 0 < k < n. In general, for each a = 2, 3, ..., if n is sufficiently large then pi(n^a + k^a) is prime for some 0 < k < n.
For any integer n > 1, it is easy to show that pi(n + k) is prime for some 0 < k < n.

			a(5) = 1 since pi(5 + 1^2) = 3 is prime.
a(6) = 1 since pi(6 + 5^2) = pi(31) = 11 is prime.
a(9) = 2 since pi(9 + 3^2) = pi(18) = 7 and pi(9 + 5^2) = pi(34) = 11 are both prime.
a(12) = 1 since pi(12 + 10^2) = pi(112) = 29 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000720, A185636, A237453, A237496, A237497, A237578.

p[n_]:=PrimeQ[PrimePi[n]]
a[n_]:=Sum[If[p[n+k^2],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

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}]

A187808 a(n) = |{0<=k

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 07 2013

nonn

Conjecture: a(n)>0 for all n>1. Moreover, if n>5 is different from 9, 191, 329, 641, 711, 979, then 2k-3, 2k+3, n(n-k)-1, n(n+k)-1 are all prime for some 0

Zhi-Wei Sun also made the following conjectures:

(1) For any integer n>101 there is an integer 0

(2) For each n=128,129,... there is an integer 0

			a(25) = 1 since 2*17+3 = 37, 25(25-17)-1 = 199, and 25(25+17)-1 = 1049 are all prime.

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

Cf. A185636, A086686, A034693.

a[n_]:=a[n]=Sum[If[PrimeQ[2k+3]==True&&PrimeQ[n(n-k)-1]==True&&PrimeQ[n(n+k)-1]==True,1,0],{k,0,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

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

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A238457 a(n) = |{0 < k <= n: p(n) + k is prime}|, where p(.) is the partition function (A000041).

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A204065 Least nonnegative integer k with n+k and n+k^2 both prime.

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A210531 Number of nonnegative integers k

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A185150 Number of odd primes p between n^2 and (n+1)^2 with (n/p) = 1, where (-) is the Legendre symbol.

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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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A237582 a(n) = |{0 < k < n: pi(n + k^2) is prime}|, where pi(.) is given by A000720.

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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.