A238134 -id:A238134 - 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}]

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}]

A238732 Number of primes p < n with floor((n-p)/3) a square.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 03 2014

nonn

Conjecture: (i) For any integers m > 2 and n > 2, there is a prime p < n such that floor((n-p)/m) is a square.
(ii) For any integer k > 2, if m and n are sufficiently large, then there is a prime p < n such that floor((n-p)/m) is a k-th power. In particular, for any integer n > 2, there is a prime p < n such that floor((n-p)/5) is a cube.
We have verified part (i) for m = 3, ..., 10 and n = 3, ..., 10^6.

			a(3) = 1 since 2 is prime with floor((3-2)/3) = 0^2.
a(11) = 1 since 7 is prime with floor((11-7)/3) = 1^2.
a(13) = 1 since 13 is prime with floor((13-11)/3) = 0^2.
a(28) = 1 since 23 is prime with floor((28-23)/3) = 1^2.
a(37) = 1 since 23 is prime with floor((37-23)/3) = 2^2.
a(173) = 1 since 97 is prime with floor((173-97)/3) = 5^2.
It seems that a(n) = 1 only for n = 3, 11, 13, 28, 37, 173.

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, A000290, A237706, A238701, A238134.

SQ[n_]:=IntegerQ[Sqrt[n]]
s[n_,k_]:=SQ[Floor[(n-Prime[k])/3]]
a[n_]:=Sum[If[s[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

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.

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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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A238732 Number of primes p < n with floor((n-p)/3) a square.

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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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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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Mathematica

A238732 Number of primes p < n with floor((n-p)/3) a square.

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Programs

Mathematica

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.