A220413 -id:A220413 - cp's OEIS frontend

A236574 Primes p with prime(p)^3 + 2p^3 and p^3 + 2prime(p)^3 both prime.

3, 79, 997, 2657, 3697, 4513, 6947, 8887, 9547, 16187, 22697, 26479, 31319, 37463, 39139, 39887, 43573, 43987, 45667, 47387, 47743, 47819, 48221, 54217, 56923, 57373, 74017, 74149, 74707, 75533, 93251, 100043

Offset: 1

Zhi-Wei Sun, Jan 29 2014

nonn

Conjecture: This sequence has infinitely many terms.

In 2001 Heath-Brown proved that there are infinitely many primes of the form x^3 + 2*y^3 with x and y positive integers.

			a(1) = 3 since prime(3)^3 + 2*3^3 = 125 + 54 = 179 and 3^3 + 2*prime(3)^3 = 27 + 2*125 = 277 are both prime, but 2^3 + 2*prime(2)^3 = 62 is composite.

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.

Cf. A000040, A000578, A173587, A220413, A236193.

p[n_]:=PrimeQ[Prime[n]^3+2*n^3]&&PrimeQ[n^3+2*Prime[n]^3]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]
Select[Prime[Range[10000]],AllTrue[{Prime[#]^3+2*#^3,#^3+2*Prime[ #]^3}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 20 2017 *)

A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 5, 4, 1, 2, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 2, 4, 3, 6, 3, 2, 4, 2, 5, 6, 4, 5, 4, 5, 5, 8, 7, 4, 7, 7, 6, 7, 4, 6, 7, 5, 6, 3, 11, 7, 1, 5, 3, 5, 6, 6, 10, 4, 13, 12, 9, 4, 9, 10, 5, 8, 3, 6, 7, 5, 4, 8, 13, 6, 3, 5, 5, 11, 6, 13, 4, 9, 10, 8, 12, 11, 8, 7, 10, 8, 7, 8, 8

Offset: 1

Zhi-Wei Sun, Dec 16 2012

nonn

Conjecture: a(n)>0 for every n=1,2,3,.... Moreover, any odd integer greater than 2092 can be written as x+y (x,y>0) with x-3, x+3 and x^18+3*y^18 all prime.
This has been verified for n up to 2*10^6.
Zhi-Wei Sun also made the following general conjecture: For each positive integer m, any sufficiently large odd integer n can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime (and hence there are infinitely many primes in the form x^m+3*y^m). In particular, for m = 1, 2, 3, 4, 5, 6, 18 any odd integer greater than one can be written as x+y (x,y>0) with x^m+3*y^m prime, and for m =1, 2, 3 any odd integer n>15 can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime.
Our computation suggests that for each m=7,...,20 any odd integer greater than 32, 10, 24, 30, 48, 36, 72, 146, 48, 48, 152, 2, 238, 84 respectively can be written as x+y (x,y>0) with x^m+3*y^m prime.

			a(3)=1 since 2*3-1=5=1+4 with 1^18+3*4^18=206158430209 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A220554, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[k^18+3*(2n-1-k)^18]==True,1,0],{k,0,2n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

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}]

A236619 a(n) = |{0 < k < n: prime(m)^3 + 2m^3 and m^3 + 2prime(m)^3 are both prime with m = 3*phi(k) + phi(n-k) - 1}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 29 2014

nonn

Conjecture: a(n) > 0 for every n = 90, 91, ....
We have verified this for n up to 100000.
The conjecture implies that there are infinitely many positive integers m with prime(m)^3 + 2*m^3 and m^3 + 2*prime(m)^3 both prime.

			a(51) = 1 since 3*phi(35) + phi(51-35) - 1 = 3*24 + 8 - 1 = 79 with prime(79)^3 + 2*79^3 = 401^3 + 2*79^3 = 65467279 and 79^3 + 2*prime(79)^3 = 79^3 + 2*401^3 = 129455441 both prime.

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

Cf. A000010, A000040, A000578, A173587, A220413, A236192, A236574.

p[n_]:=PrimeQ[Prime[n]^3+2*n^3]&&PrimeQ[n^3+2*Prime[n]^3]
f[n_,k_]:=3*EulerPhi[k]+EulerPhi[n-k]-1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A344173 Number of primes p < prime(n) of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that p is a quadratic nonresidue modulo prime(n).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, May 10 2021

nonn

In 2001 Heath-Brown proved that there are infinitely many primes of the form x^3 + 2*y^3 with x and y nonnegative integers.
Conjecture: (i) a(n) > 0 for all n > 1. In other words, for each odd prime p, there is a prime q < p of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a quadratic nonresidue modulo p.
(ii) For any prime p > 5 not equal to 29, there is a prime q < p of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a quadratic residue modulo p.
Part (i) of the conjecture verified for all odd primes p < 2*10^9.
We even conjecture further that for any prime p > 5 there is a prime q < p of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a primitive root modulo p.
See also A344174 for a similar conjecture.

			a(2) = 1, and the prime 0^3 + 2*(2-1)^3 = 2 is a quadratic nonresidue modulo prime(2) = 3.
a(6) = 1, and the prime 0^3 + 2*(2-1)^3 = 2 is a quadratic nonresidue modulo prime(6) = 13.
a(20) = 1, and the prime 1^3 + 2*(2-1)^3 = 17 is a quadratic nonresidue modulo prime(20) = 71.
a(48) = 1, and the prime 1^3 + 2*(2-1)^3 = 3 is a quadratic nonresidue modulo prime(48) = 223.
a(88) = 1, and the prime 3^3 + 2*(3-1)^3 = 43 is a quadratic nonresidue modulo prime(88) = 457.

D. R. Heath-Brown, Primes represented by x^3 + 2y^3, Acta Mathematica 186 (2001), 1-84.

Cf. A000040, A000578, A173587, A220413, A344174.

tab={0};Do[p:=p=Prime[n];tt={};Do[If[PrimeQ[b+1]&&PrimeQ[a^3+2b^3]&&JacobiSymbol[a^3+2b^3,p]==-1,tt=Append[tt,a^3+2b^3]],{a,0,(p-1)^(1/3)},{b,1,((p-1-a^3)/2)^(1/3)}];tab=Append[tab,Length[Union[tt]]],{n,2,100}];Print[tab]

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A236574 Primes p with prime(p)^3 + 2*p^3 and p^3 + 2*prime(p)^3 both prime.

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A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.

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A224030 a(n) = |{0

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A236619 a(n) = |{0 < k < n: prime(m)^3 + 2*m^3 and m^3 + 2*prime(m)^3 are both prime with m = 3*phi(k) + phi(n-k) - 1}|, where phi(.) is Euler's totient function.

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A344173 Number of primes p < prime(n) of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that p is a quadratic nonresidue modulo prime(n).

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A236574 Primes p with prime(p)^3 + 2p^3 and p^3 + 2prime(p)^3 both prime.

A236619 a(n) = |{0 < k < n: prime(m)^3 + 2m^3 and m^3 + 2prime(m)^3 are both prime with m = 3*phi(k) + phi(n-k) - 1}|, where phi(.) is Euler's totient function.