A238878 -id:A238878 - cp's OEIS frontend

A238881 Number of odd primes p < 2n with prime(n(p+1)/2) + n*(p+1)/2 prime.

0, 1, 1, 0, 0, 2, 1, 3, 1, 2, 2, 5, 2, 3, 2, 4, 3, 5, 2, 3, 2, 4, 1, 8, 4, 4, 4, 3, 2, 5, 2, 4, 4, 4, 4, 8, 2, 4, 2, 5, 4, 6, 3, 2, 6, 6, 3, 11, 6, 10, 4, 8, 2, 11, 4, 7, 4, 7, 2, 12, 4, 6, 2, 6, 3, 8, 3, 5, 8, 12, 6, 12, 4, 15, 8, 11, 5, 12, 2, 11

Offset: 1

Zhi-Wei Sun, Mar 06 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5, and a(n) = 1 only for n = 2, 3, 7, 9, 23. Moreover, for any r = 1,-1 and n > 5*(2+r) there is a positive integer k < n such that 2*k+r and prime(k*n)+k*n are both prime.
(ii) If n > 1 is not equal to 13, then prime(k*n) - k*n is prime for some k = 1, ..., n.
This conjecture implies that there are infinitely many positive integers m with prime(m) + m (or prime(m) - m) prime.

			a(7) = 1 since 11 and prime(7*(11+1)/2) + 7*(11+1)/2 = prime(42) + 42 = 181 + 42 = 223 are both prime.
a(23) = 1 since 7 and prime(23*(7+1)/2) + 23*(7+1)/2 = prime(92) + 92 = 479 + 92 = 571 are both 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. (See Conjecture 3.21(i) and note that the typo 2k+1 there should be 2k-1.)

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A064269, A064402, A238573, A238576, A238878.

PQ[n_]:=PrimeQ[Prime[n]+n]
p[k_,n_]:=PQ[(Prime[k]+1)/2*n]
a[n_]:=Sum[If[p[k,n],1,0],{k,2,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]

a(n) = {my(nb = 0); forprime(p=3, 2*n, if (isprime(prime(n*(p+1)/2) + n*(p+1)/2), nb++);); nb;} \\ Michel Marcus, Sep 21 2015

A238890 a(n) = |{0 < k <= n: prime(kn) - pi(kn) is prime}|, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 06 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 28.
(ii) If n > 7 is not equal to 34, then prime(k*n) + pi(k*n) is prime for some k = 1, ..., n.
The conjecture implies that there are infinitely many primes p with p - pi(pi(p)) (or p + pi(pi(p))) prime.

			a(5) = 1 since prime(3*5) - pi(3*5) = 47 - 6 = 41 is prime.
a(28) = 1 since prime(18*28) - pi(18*28) = prime(504) - pi(504) = 3607 - 96 = 3511 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A000720, A237578, A237712, A238573, A238576, A238878, A238881.

p[k_]:=PrimeQ[Prime[k]-PrimePi[k]]
a[n_]:=Sum[If[p[k*n],1,0],{k,1,n}]
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}]

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A238881 Number of odd primes p < 2n with prime(n(p+1)/2) + n*(p+1)/2 prime.

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A238890 a(n) = |{0 < k <= n: prime(kn) - pi(kn) is prime}|, where pi(x) denotes the number of primes not exceeding x.

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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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cp's OEIS Frontend

A238881 Number of odd primes p < 2*n with prime(n*(p+1)/2) + n*(p+1)/2 prime.

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A238890 a(n) = |{0 < k <= n: prime(k*n) - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.

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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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A238881 Number of odd primes p < 2n with prime(n(p+1)/2) + n*(p+1)/2 prime.

A238890 a(n) = |{0 < k <= n: prime(kn) - pi(kn) is prime}|, where pi(x) denotes the number of primes not exceeding x.