A173587 -id:A173587 - cp's OEIS frontend

A220419 Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 all prime.

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

Offset: 1

Zhi-Wei Sun, Dec 14 2012

nonn

Conjecture: a(n)>0 for all n>527.
This has been verified for n up to 2*10^7. It implies the Goldbach conjecture since 2(x+y)=(2x+1)+(2y-1).
Zhi-Wei Sun also made the following similar conjectures:
(1) Each integer n>1544 can be written as x+y (x>0, y>0) with 2x-1, 2y+1 and x^3+2y^3 all prime.
(2) Any odd number n>2060 can be written as 2p+q with p, q and p^3+2((q-1)/2)^3 all prime.
(3) Every integer n>25537 can be written as p+q (q>0) with p, p-6, p+6 and p^3+2q^3 all prime.
(4) Any even number n>1194 can be written as x+y (x>0, y>0) with x^3+2y^3 and 2x^3+y^3 both prime.
(5) Each integer n>3662 can be written as x+y (x>0, y>0) with 3(xy)^3-1 and 3(xy)^3+1 both prime.
(6) Any integer n>22 can be written as x+y (x>0, y>0) with (xy)^4+1 prime. Also, any integer n>7425 can be written as x+y (x>0, y>0) with 2(xy)^4-1 and 2(xy)^4+1 both prime.
(7) Every odd integer n>1 can be written as x+y (x>0, y>0) with x^4+y^2 prime. Moreover, any odd number n>15050 can be written as p+2q with p, q and p^4+(2q)^2 all prime.
Conjectures (1) to (7) verified up to 10^6. - Mauro Fiorentini, Sep 22 2023

			a(25)=1 since 25=3+22 with 2*3+1, 2*22-1 and 3^3+2*22^3=21323 all prime.
a(26)=1 since 26=11+15 with 2*11+1, 2*15-1 and 11^3+2*15^3=8081 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
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. A220413, A173587, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[2k+1]==True&&PrimeQ[2(n-k)-1]==True&&PrimeQ[k^3+2(n-k)^3]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]

A220431 Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 6, 1, 3, 6, 4, 3, 3, 2, 3, 4, 3, 4, 2, 3, 3, 5, 4, 4, 7, 1, 2, 5, 1, 5, 7, 4, 2, 3, 7, 4, 7, 2, 4, 7, 4, 4, 5, 2, 5, 8, 4, 3, 3, 5, 2, 8, 5, 4, 3, 10, 7, 8, 2, 3, 5, 5, 3, 6, 3, 3, 14, 4, 3, 12, 3, 7, 7, 5, 6, 8, 7, 5, 9, 9, 4, 4, 3, 6, 10, 8

Offset: 1

Zhi-Wei Sun, Dec 14 2012

nonn

Conjecture: a(n)>0 for all n>3.
This has been verified for n up to 10^8, and it is stronger than A. Murthy's conjecture related to A109909.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023
The conjecture implies the twin prime conjecture for the following reason: If x_1<...Zhi-Wei Sun also made some similar conjectures. For example, any integer n>2 not equal to 63 can be written as x+y (x>0, y>0) with 2x-1, 2x+1 and 2xy+1 all prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Jul 26 2023

			a(22)=1 since 22=4+18 with 3*4-1, 3*4+1 and 4*18-1 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. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[3k-1]==True&&PrimeQ[3k+1]==True&&PrimeQ[k(n-k)-1]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]

A219726 Integers of the form x^3 + 2y^3 (x, y > 0).

Original entry on oeis.org

3, 10, 17, 24, 29, 43, 55, 62, 66, 80, 81, 118, 127, 129, 136, 141, 155, 179, 192, 218, 232, 251, 253, 258, 270, 277, 314, 344, 345, 359, 375, 397, 433, 440, 459, 466, 471, 496, 514, 528, 557, 566, 593, 640, 648, 687, 694, 713, 731, 745, 750, 762, 775, 783

Offset: 1

Zak Seidov, Nov 26 2012

nonn

D. R. Heath-Brown proved in 2001 that there are infinitely many prime numbers in this sequence. These primes are in A173587. - Bernard Schott, Apr 07 2020

Charles R Greathouse IV, 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.
Wikipedia, Roger Heath-Brown

Cf. A173587, A219722, A219725.

m = 10^3; Union[Flatten@Table[x^3 + 2 y^3, {x, m^(1/3)}, {y, ((m - x^3)/2)^(1/3)}]]

is(n)=for(y=1,sqrtnint((n-1)\2,3), if(ispower(n-2*y^3,3),return(1)));0 \\ Charles R Greathouse IV, Apr 07 2020

list(lim)=my(v=List(),Y); lim\=1; for(y=1,sqrtnint((lim-1)\2,3), Y=2*y^3; for(x=1,sqrtnint(lim-Y,3), listput(v,x^3+Y))); Set(v) \\ Charles R Greathouse IV, Apr 07 2020

A219728 Squares of the form x^3 + 2*y^3, with x, y > 0.

Original entry on oeis.org

81, 5041, 5184, 10000, 46225, 59049, 77841, 83521, 322624, 331776, 640000, 685584, 707281, 1265625, 2958400, 3157729, 3418801, 3674889, 3779136, 4157521, 4981824, 5345344, 5602689, 5736025, 7290000, 9150625, 9529569, 10725625, 12257001, 14776336, 15904144

Offset: 1

Zak Seidov, Nov 26 2012

nonn

Subsequence of A219726.

			81: (x = 3, y = 3), 5041: (x = 17,  y = 4).

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A173587, A219722, A219725, A219726.

With[{nn=300},Select[Flatten[Table[x^3+2y^3,{x,nn},{y,nn}]],IntegerQ[ Sqrt[#]]&]//Union] (* Harvey P. Dale, Oct 24 2018 *)

A199800 Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 21 2012

nonn

Conjecture: a(n)>0 for all n>11.
This implies the twin prime conjecture, and it has been verified for n up to 10^9.
Zhi-Wei Sun also made some similar conjectures, for example, any integer n>5 can be written as p+q with p, 2q-3 and 2q+3 all prime, and each integer n>4 can be written as p+q with p, 3q-2+(n mod 2) and 3q+2-(n mod 2) all prime.

			a(3)=1 since 3=2+1 with 2, 6*1-1 and 6*1+1 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. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[n-k]==True&&PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

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

Original entry on oeis.org

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 *)

A212287 Primes of the form m*p^2 + 1, where p is prime and m <= p^2.

Original entry on oeis.org

5, 13, 17, 19, 37, 73, 101, 151, 197, 251, 401, 491, 601, 677, 727, 883, 1373, 1453, 1471, 1667, 2029, 2179, 2663, 3389, 3469, 3631, 3719, 4057, 4357, 4733, 5477, 6359, 6761, 7019, 8093, 8713, 8837, 9127, 9439, 9803, 9923, 10093, 10141, 10831, 10891, 11617, 11831, 12101, 12343

Offset: 1

Charles R Greathouse IV, Jun 13 2012

nonn

Not known to be infinite, but see the Matomäki link.

			13 is a member since 13 = 3 * 2^2 + 1 with 3 <= 2^2 and 3 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, A note on primes of the form p = aq^2 + 1, Acta Arith. 137 (2009), pp. 133-137.

Cf. A028916, A173587.

list(lim)=my(v=List(),t);lim=lim\1-.5;forprime(p=2,sqrt(lim), for(a=1,min(lim\p^2,p^2),if(isprime(t=a*p^2+1),listput(v,t))));vecsort(Vec(v),,8)

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}]

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cp's OEIS Frontend

A219722 Primes expressible as x^3 + 2*y^3 (x, y > 0) in two ways.

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A220419 Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 all prime.

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A219725 Integers expressible as x^3 + 2*y^3 (x, y > 0) in two ways.

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A220431 Number of ways to write n=x+y (x>0, y>0) with 3x-1, 3x+1 and xy-1 all prime.

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A219726 Integers of the form x^3 + 2y^3 (x, y > 0).

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A219728 Squares of the form x^3 + 2*y^3, with x, y > 0.

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A199800 Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime.

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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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A212287 Primes of the form m*p^2 + 1, where p is prime and m <= p^2.

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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.