A236832 -id:A236832 - cp's OEIS frontend

A238766 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that prime(prime(k)) - prime(k) + 1, prime(prime(2k+1)) - prime(2k+1) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.

0, 1, 1, 2, 3, 2, 4, 3, 2, 4, 1, 4, 3, 4, 6, 3, 6, 3, 3, 4, 3, 3, 2, 6, 4, 4, 5, 3, 3, 5, 4, 4, 4, 3, 4, 3, 6, 5, 2, 6, 3, 4, 6, 1, 3, 3, 6, 4, 6, 6, 4, 4, 5, 5, 1, 5, 3, 3, 6, 5, 6, 4, 7, 6, 8, 6, 8, 3, 9, 8, 9, 10, 8, 11, 6, 10, 10, 4, 5, 4

Offset: 1

Zhi-Wei Sun, Mar 05 2014

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 11, 44, 55, 149, 371.

This suggests that there are infinitely many prime pairs {p, q} with 2*pi(p) + 1 = pi(q) such that prime(p) - p + 1 and prime(q) - q + 1 are both prime.

			a(3) = 1 since 3 = 1 + 2 with prime(prime(1)) - prime(1) + 1 = prime(2) - 2 + 1 = 2, prime(prime(2*1+1)) - prime(2*1+1) + 1 = prime(5) - 5 + 1 = 7 and prime(prime(2)) - prime(2) + 1 = prime(3) - 3 + 1 = 3 all prime.
a(371) = 1 since 371 = 66 + 305 with prime(prime(66)) - prime(66) + 1 = prime(317) - 317 + 1 = 2099 - 316 = 1783, prime(prime(2*66+1)) - prime(2*66+1) + 1 = prime(751) - 751 + 1 = 5701 - 750 = 4951 and prime(prime(305)) - prime(305) + 1 = prime(2011) - 2011 + 1 = 17483 - 2010 = 15473 all 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. A000040, A234694, A234695, A235189, A236832, A238134, A238756, A238776.

pq[k_]:=PrimeQ[Prime[Prime[k]]-Prime[k]+1]
a[n_]:=Sum[If[pq[k]&&pq[2k+1]&&pq[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A237291 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q, r, pi(p), pi(q), pi(r) all prime, where pi(x) denotes the number of primes not exceeding x (A000720).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 06 2014

nonn

Conjecture: a(n) > 0 for all n > 36.

This is stronger than Goldbach's weak conjecture finally proved by H. A. Helfgott in 2013.

			a(16) = 1 since 2*16 - 1 = 3 + 11 + 17 with 3, 11, 17, pi(3) = 2, pi(11) = 5 and pi(17) = 7 all prime.
a(179) = 1 since 2*179 - 1 = 83 + 83 + 191 with 83, 191, pi(83) = 23 and pi(191) = 43 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252, 2012.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897, 2013.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000720, A006450, A068307, A230219, A236832, A237284.

p[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
a[n_]:=Sum[If[p[2n-1-Prime[Prime[i]]-Prime[Prime[j]]],1,0],{i,1,PrimePi[PrimePi[(2n-1)/3]]},{j,i,PrimePi[PrimePi[(2n-1-Prime[Prime[i]])/2]]}]
Table[a[n],{n,1,80}]

A238756 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that 2*k + 1, prime(prime(k)) - prime(k) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 05 2014

nonn

Conjecture: a(n) > 0 for all n > 1.
We have verified this for n up to 10^7.
The conjecture suggests that there are infinitely many primes p with 2*pi(p) + 1 and prime(p) - p + 1 both prime.

			a(6) = 2 since 6 = 2 + 4 with 2*2 + 1 = 5, prime(prime(2)) - prime(2) + 1 = prime(3) - 3 + 1 = 3 and prime(prime(4)) - prime(4) + 1 = prime(7) - 7 + 1 = 17 - 6 = 11 all prime, and 6 = 3 + 3 with 2*3 + 1 = 7 and prime(prime(3)) - prime(3) + 1 = prime(5) - 5 + 1 = 11 - 4 = 7 both 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. A000040, A218829, A234694, A234695, A235189, A236832, A237413, A238134.

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

A260753 Least positive integer k such that both k and k*n belong to the set {m>0: prime(prime(m))-prime(m)+1 = prime(p) for some prime p}.

Original entry on oeis.org

2, 2, 2279, 5806, 4, 1135, 816, 6556, 725, 2, 1333, 10839, 27, 829, 2279, 2838, 3881, 6540, 2564, 2, 7830, 6540, 27, 4905, 6121, 8220, 316, 1061, 2, 14691, 2, 1168, 738, 4707, 785, 12467, 5492, 1447, 542, 538, 12840, 829, 4732, 5637, 785, 1246, 1198, 433, 58, 573, 26280, 17387, 316, 430, 1198, 4315, 4315, 1479, 4315, 1497

Offset: 1

Zhi-Wei Sun, Aug 18 2015

nonn

Conjecture: For any s and t in the set {1,-1}, every positive rational number r can be written as m/n with m and n in the set {k>0: prime(prime(k))+s*prime(k)+t = prime(p) for some prime p}.

In the case s = -1 and t = 1, the conjecture implies that A261136 has infinitely many terms.

			a(3) = 2279 since  prime(prime(2279))-prime(2279)+1 = prime(20147)-20147+1 = 226553-20146 = 206407 = prime(18503) with 18503 prime, and  prime(prime(2279*3))-prime(2279*3)+1 = prime(68777)-68777+1 = 865757-68776 = 796981 = prime(63737) with 63737 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..5000
Zhi-Wei Sun, Checking the conjecture for s = -1, t = 1 and r = a/b (a,b = 1..125)
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A234694, A234695, A236832, A238766, A238878, A261136, A261362.

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

cp's OEIS Frontend

A238766 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that prime(prime(k)) - prime(k) + 1, prime(prime(2*k+1)) - prime(2*k+1) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.

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A237291 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q, r, pi(p), pi(q), pi(r) all prime, where pi(x) denotes the number of primes not exceeding x (A000720).

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A238756 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that 2*k + 1, prime(prime(k)) - prime(k) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.

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A260753 Least positive integer k such that both k and k*n belong to the set {m>0: prime(prime(m))-prime(m)+1 = prime(p) for some prime p}.

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A238766 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that prime(prime(k)) - prime(k) + 1, prime(prime(2k+1)) - prime(2k+1) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.