A046927 -id:A046927 - cp's OEIS frontend

A218754 Number of ways to write n=p+q(3+(-1)^n)/2 with q<=n/2 and p, q, p^2+3pq+q^2 all prime.

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

Offset: 1

Zhi-Wei Sun, Nov 04 2012

nonn

Conjecture: a(n)>0 for all n>=1188.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 23 2023
This conjecture is stronger than both Goldbach's conjecture and Lemoine's conjecture.
Zhi-Wei Sun also made the following conjecture: Given any positive odd integer d, there is a prime p(d) such that for any prime p>p(d) there is a prime q
Conjecture verified for d up to 100 and p up to 10^7. - Mauro Fiorentini, Sep 23 2023

			For n=72 we have a(72)=1 since the only primes p and q with p+q=72, q<=36 and p^2+3pq+q^2 prime are p=67 and q=5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A002372, A046927, A218585, A218654, A218656.

Cf. A000034 = 1,2,1,2,... = (3-(-1)^n)/2. (Note: Offset shifted w.r.t. use in the definition of this sequence.) - M. F. Hasler, Nov 05 2012

a[n_]:=a[n]=Sum[If[PrimeQ[q]==True&&PrimeQ[n-q(3-(-1)^n)/2]&&PrimeQ[q^2+3q(n-q(3-(-1)^n)/2)+(n-q(3-(-1)^n)/2)^2]==True,1,0],{q,1,n/2}]
Do[Print[n," ",a[n]],{n,1,20000}]

A218825 Number of ways to write 2n-1 as p+2q with p, q and p^2+60q^2 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Nov 07 2012

nonn

Conjecture: a(n)>0 for all n>8.
This conjecture is stronger than Lemoine's conjecture. It has been verified for n up to 10^8.
Conjecture verified for 2n-1 up to 10^9. - Mauro Fiorentini, Jul 20 2023
Zhi-Wei Sun also made the following general conjecture: For any positive integer n, the set E(n) of positive odd integers not of the form p+2q with p, q, p^2+4(2^n-1)q^2 all prime, is finite. In particular, if we let M(n) denote the maximal element of E(n), then M(1)=3449, M(2)=1711, E(3)={1,3,5,7,31,73}, E(4)={1,3,5,7,9,11,13,15},
  M(5)=6227, M(6)=1051, M(7)=2239, M(8)=2599, M(9)=7723,
  M(10)=781, M(11)=1163, M(12)=587, M(13)=11443,
  M(14)=2279, M(15)=157, M(16)=587, M(17)=32041,
  M(18)=1051, M(19)=2083, M(20)=4681.
Conjecture verified for 2n-1 up to 10^9 for n <= 4 and up to 10^6 for n <= 20. - Mauro Fiorentini, Jul 20 2023
Zhi-Wei Sun also guessed that for any positive even integer d not congruent to 2 modulo 6 there exists a prime p(d) such that for any prime p>p(d) there is a prime q
  p(4)=p(6)=3, p(10)=5, p(12)=3, p(16)=2, p(18)=3,
  p(22)=11, p(24)=17, and p(28)=p(30)=7.

			a(10)=1 since the only primes p and q with p^2+60q^2 prime and p+2q=19 are p=13 and q=3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.
Wikipedia, Lemoine's conjecture

Cf. A000040, A046927, A218754, A218585, A218654, A218656, A218797.

a[n_]:=a[n]=Sum[If[PrimeQ[q]==True&&PrimeQ[2n-1-2q]==True&&PrimeQ[(2n-1-2q)^2+60q^2]==True,1,0],{q,1,n-1}]
Do[Print[n," ",a[n]],{n,1,20000}]

A218825(n)={my(c=0,n21=n*2-1);forprime(q=2,n-1,isprime(n21-2*q) || next; isprime(q^2*60+(n21-2*q)^2) && c++); c}  \\ M. F. Hasler, Nov 07 2012

A218867 Number of prime pairs {p,q} with p>q and {p-4,q+4} also prime such that p+(1+(n mod 6))q=n if n is not congruent to 4 (mod 6), and p-q=n and q

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Nov 13 2012

nonn

Conjecture: a(n)>0 for all n>50000 with n different from 50627, 61127, 66503.

This conjecture implies that there are infinitely many cousin prime pairs. It is similar to the conjectures related to A219157 and A219055.

			a(20)=1 since 20=11+3*3 with 11-4 and 3+4 prime. a(28)=1 since 28=41-13 with 41-4 and 13+4 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.
Zhi-Wei Sun, Table of n, a(n) for n = 1..10^5.

Cf. A023200, A046132, A219157, A219055, A002375, A046927, A218754, A218825, A219052.

c[n_]:=c[n]=If[Mod[n+2,6]==0,1,-1-Mod[n,6]]; d[n_]:=d[n]=2+If[Mod[n+2,6]>0,Mod[n,6],0]; a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+4] == True && PrimeQ[n+c[n]Prime[k]] == True && PrimeQ[n+c[n]Prime[k]-4]==True,1,0], {k,1,PrimePi[(n-1)/d[n]]}]; Do[Print[n," ",a[n]], {n,100}]

A219157 Number of prime pairs {p,q} with p>q and p-2,q+2 also prime such that p+(1+mod(-n,6))q=n if n is not congruent to 2 mod 6, and p-q=n and q

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 0, 2, 0, 2, 2, 1, 1, 2, 3, 1, 0, 2, 1, 1, 0, 2, 2, 1, 2, 1, 2, 1, 0, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 0, 1, 3, 1, 0

Offset: 1

Zhi-Wei Sun, Nov 12 2012

nonn

Conjecture: a(n)>0 for all n>30000 with n different from 38451, 46441, 50671, 62371.

This conjecture is stronger than the twin prime conjecture. It is similar to the conjecture associated with A219055 about sexy prime pairs.

			a(16)=1 since 16=7+3*3 with 7-2 and 3+2 prime. a(26)=1 since 26=31-5 with 31-2 and 5+2 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A002375, A046927, A219055, A218754, A218825, A219052.

c[n_]:=c[n]=If[Mod[n-2,6]==0,1,-1-Mod[-n,6]]
d[n_]:=d[n]=2+If[Mod[n-2,6]>0,Mod[-n,6],0]
a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[n+c[n]Prime[k]]==True&&PrimeQ[n+c[n]Prime[k]-2]==True,1,0],
{k,1,PrimePi[(n-1)/d[n]]}]
Do[Print[n," ",a[n]],{n,1,100000}]

A219185 Number of prime pairs {p,q} (p>q) with 3(p-q)-1 and 3(p-q)+1 both prime such that p+(1+(n mod 2))q=n.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Nov 13 2012

nonn

Conjecture: a(n)>0 for all odd n>4676 and even n>30986.

This conjecture has been verified for n up to 5*10^7. It implies Goldbach's conjecture, Lemoine's conjecture and the twin prime conjecture.

			a(11)=1 since 11=5+2*3, and both 3(5-3)-1=5 and 3(5-3)+1=7 are prime.
a(16)=2 since 16=11+5=13+3, and 3(11-5)-1, 3(11-5)+1, 3(13-3)-1, 3(13-3)+1 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. A001359, A006512, A002375, A046927, A219157, A219055, A218867, A218754, A218825, A219052.

a[n_]:=a[n]=Sum[If[PrimeQ[n-(1+Mod[n,2])Prime[k]]==True&&PrimeQ[3(n-(2+Mod[n,2])Prime[k])-1]==True&&PrimeQ[3(n-(2+Mod[n,2])Prime[k])+1]==True,1,0],
{k,1,PrimePi[(n-1)/(2+Mod[n,2])]}]
Do[Print[n," ",a[n]],{n,1,100000}]

a(n)=if(n%2, aOdd(n), aEven(n))
aOdd(n)=my(s); forprime(q=2,(n-1)\3, my(p=n-2*q); if(isprime(n-2*q) && isprime(3*n-9*q-1) && isprime(3*n-9*q+1), s++)); s
aEven(n)=my(s); forprime(q=2,n/2, if(isprime(n-q) && isprime(3*n-6*q-1) && isprime(3*n-6*q+1), s++)); s
\\ Charles R Greathouse IV, Jul 31 2016

A209315 Number of ways to write 2n-1 = p+q with q practical, p and q-p both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 19 2013

nonn

Conjecture: a(n)>0 for all n>8.
This has been verified for n up to 10^7.
As p+q=2p+(q-p), the conjecture implies Lemoine's conjecture related to A046927.
Zhi-Wei Sun also conjectured that any integer n>2 can be written as p+q, where p is a prime,  one of q and q+1 is prime and another of q and q+1 is practical.

			a(9)=1 since 2*9-1=5+12 with 12 practical, 5 and 12-5 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
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, A046927, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185.

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[PrimeQ[p]==True&&pr[2n-1-p]==True&&PrimeQ[2n-1-2p]==True,1,0],{p,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A235189 Number of ways to write n = (1 + (n mod 2))*p + q with p < n/2 such that p, q and prime(p) - p + 1 are all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 04 2014

nonn

Conjecture: a(n) > 0 for all n > 6.

This implies both Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927). For primes p with prime(p) - p + 1 also prime, see A234695.

			a(10) = 1 since 10 = 3 + 7 with 3, 7 and prime(3) - 3 + 1 = 3 all prime.
a(28) = 1 since 28 = 5 + 23 with 5, 23 and prime(5) - 4 = 7 all prime.
a(61) = 1 since 61 = 2*7 + 47 with 7, 47 and prime(7) - 6 = 11 all prime.
a(98) = 1 since 98 = 31 + 67 with 31, 67 and prime(31) - 30 = 97 all prime.

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

Cf. A000040, A045917, A046927, A234694, A234695, A235187.

p[n_]:=PrimeQ[Prime[n]-n+1]
a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[n-(1+Mod[n,2])*Prime[k]],1,0],{k,1,PrimePi[(n-1)/2]}]
Table[a[n],{n,1,100}]

A236566 Number of ordered ways to write 2*n = p + q with p, q and prime(p + 2) + 2 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 28 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2.
(ii) If n > 30, then 2*n + 1 can be written as 2*p + q with p, q and prime(p + 2) + 2 all prime.
Part (i) implies both the Goldbach conjecture and the twin prime conjecture. If all primes p with prime(p + 2) + 2 are smaller than an even number N > 2, then for any such a prime p the number N! + N - p is in the interval (N!, N! + N) and hence not prime.
Similarly, part (ii) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture.
We have verified part (i) of the conjecture for n up to 2*10^8.

			a(10) = 1 since 2*10 = 3 + 17 with 3, 17 and prime(3 + 2) + 2 = 11 + 2 = 13 all prime.
a(589) = 1 since 2*589 = 577 + 601 with 577, 601 and prime(577 + 2) + 2 = 4229 + 2 = 4231 all prime.

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

Cf. A000040, A001359, A002372, A002375, A006512, A046927, A236531.

p[m_]:=PrimeQ[Prime[m+2]+2]
a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[2n-Prime[k]],1,0],{k,1,PrimePi[2n-1]}]
Table[a[n],{n,1,100}]

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A219864 Number of ways to write n as p+q with p and 2pq+1 both prime.

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A219055 Number of ways to write n = p+q(3-(-1)^n)/2 with p>q and p, q, p-6, q+6 all prime.

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A218754 Number of ways to write n=p+q(3+(-1)^n)/2 with q<=n/2 and p, q, p^2+3pq+q^2 all prime.

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A218825 Number of ways to write 2n-1 as p+2q with p, q and p^2+60q^2 all prime.

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A218867 Number of prime pairs {p,q} with p>q and {p-4,q+4} also prime such that p+(1+(n mod 6))q=n if n is not congruent to 4 (mod 6), and p-q=n and q

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A219157 Number of prime pairs {p,q} with p>q and p-2,q+2 also prime such that p+(1+mod(-n,6))q=n if n is not congruent to 2 mod 6, and p-q=n and q

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A219185 Number of prime pairs {p,q} (p>q) with 3(p-q)-1 and 3(p-q)+1 both prime such that p+(1+(n mod 2))q=n.

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A209315 Number of ways to write 2n-1 = p+q with q practical, p and q-p both prime.

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