A232465 -id:A232465 - cp's OEIS frontend

A237496 Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m) - 2 prime, where pi(.) is given by A000720.

0, 0, 0, 0, 0, 1, 2, 4, 4, 3, 2, 3, 3, 3, 5, 3, 1, 4, 5, 5, 7, 4, 1, 2, 1, 1, 1, 1, 1, 3, 6, 7, 8, 8, 8, 8, 8, 9, 11, 11, 11, 11, 9, 7, 7, 4, 1, 2, 1, 2, 3, 5, 7, 10, 14, 14, 14, 10, 6, 10, 14, 16, 19, 16, 13, 12, 11, 10, 7, 6, 5, 3, 3, 4, 3, 6, 9, 13, 17, 18

Offset: 1

Zhi-Wei Sun, Feb 08 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) Any integer n > 23 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) prime. Also, each integer n > 25 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) - 1 prime.

			a(6) = 1 since 6 = 3 + 3 with pi(3) + pi(3) - 2 = 2 + 2 - 2 = 2 prime.
a(17) = 1 since 17 = 2 + 15 with pi(2) + pi(15) - 2 = 1 + 6 - 2 = 5 prime.
a(99) = 1 since 99 = 1 + 98 with pi(1) + pi(98) - 2 = 0 + 25 - 2 = 23 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Cf. A000040, A000720, A232465, A237284, A237291, A237453, A237497.

PQ[n_]:=n>0&&PrimeQ[n]
p[k_,m_]:=PQ[PrimePi[k]+PrimePi[m]-2]
a[n_]:=Sum[If[p[k,n-k],1,0],{k,1,n/2}]
Table[a[n],{n,1,80}]

A232502 Number of ways to write n = k + m (0 < k < m) with 2*prime(m) - prime(k) prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 2, 3, 1, 5, 2, 3, 1, 4, 5, 3, 4, 2, 3, 2, 3, 5, 5, 3, 7, 1, 5, 4, 8, 3, 4, 5, 6, 5, 1, 6, 4, 9, 3, 8, 4, 6, 3, 10, 5, 8, 4, 8, 3, 9, 6, 4, 4, 4, 10, 6, 10, 4, 11, 5, 11, 6, 9, 5, 10, 9, 8, 6, 9, 7, 9, 11, 9, 11, 5, 10, 9, 12, 6, 6, 10, 9, 8, 13, 4, 12, 10, 12, 8, 7, 12, 14

Offset: 1

Zhi-Wei Sun, Nov 24 2013

nonn

Note that prime(k), prime(m), 2*prime(m) - prime(k) form a three-term arithmetic progression. It is known that there are infinitely many nontrivial three-term arithmetic progressions whose terms are all prime.
Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 6, 7, 9, 11, 14, 17, 21, 34, 43.
(ii) Any integer n > 4, can be written as k + m (0 < k < m) with 2*prime(m) + prime(k) prime.

			a(17) = 1 since 2*prime(10) - prime(7) = 2*29 - 17 = 41 is prime.
a(21) = 1 since 2*prime(19) - prime(2) = 2*67 - 3 = 131 is prime.
a(34) = 1 since 2*prime(24) - prime(10) = 2*89 - 29 = 149 is prime.
a(43) = 1 since 2*prime(28) - prime(15) = 2*107 - 47 = 167 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167(2008), 481-547.
J. G. van der Corput, Über Summen von Primzahlen und Primzahlquadraten, Math. Ann. 116 (1939), 1-50.

Cf. A000040, A232463, A232465.

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

A233296 a(n) = |{0 < k < n: k*prime(n-k) + 1 is prime}|.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 4, 2, 2, 5, 4, 1, 6, 1, 2, 5, 4, 4, 4, 3, 3, 2, 4, 6, 5, 4, 5, 6, 7, 7, 6, 5, 8, 6, 4, 5, 7, 8, 4, 6, 7, 6, 10, 7, 4, 8, 11, 9, 11, 6, 5, 5, 8, 8, 10, 5, 7, 10, 10, 11, 7, 6, 6, 12, 6, 10, 11, 6, 11, 7, 11, 8, 12, 7, 7, 9, 13, 9, 10, 14, 7, 13, 8, 10, 11, 9, 14, 10, 14, 17, 14, 13, 8, 12, 12

Offset: 1

Zhi-Wei Sun, Dec 07 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Similarly, for any integer n > 2, k*prime(n-k) - 1 (or k^2*prime(n-k) - 1) is prime for some 0 < k < n.

(ii) Let n > 3 be an integer. Then k + prime(n-k) is prime for some 0 < k < n. Also, if n is not equal to 13, then k^2 + prime(n-k)^2 is prime for some 0 < k < n.

			a(17) = 1 since 17 = 14 + 3 with 14*prime(3) + 1 = 14*5 + 1 = 71 prime.
a(19) = 1 since 19 = 18 + 1 with 18*prime(1) + 1 = 18*2 + 1 = 37 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A232465, A232502, A233150, A233183, A233204, A233206.

a[n_]:=Sum[If[PrimeQ[k*Prime[n-k]+1],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A233439 a(n) = |{0 < k < n: prime(k)^2 + 4*prime(n-k)^2 is prime}|.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 2, 1, 3, 4, 4, 8, 4, 6, 3, 1, 7, 3, 8, 5, 2, 9, 2, 11, 8, 7, 5, 4, 8, 7, 8, 8, 8, 7, 5, 9, 5, 10, 9, 7, 13, 9, 11, 10, 14, 5, 11, 10, 10, 11, 12, 7, 13, 10, 10, 8, 15, 11, 12, 11, 13, 14, 6, 12, 11, 22, 21, 5, 15, 7, 13, 15, 17, 15, 10, 16, 11, 13, 14, 12, 17, 12, 16, 16, 19, 22, 17, 12, 19, 17, 19, 17, 16, 17, 18, 20, 19, 17, 10, 16

Offset: 1

Zhi-Wei Sun, Dec 09 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) For any integer n > 10, prime(j)^3 + 2*prime(n-j)^2 is prime for some 0 < j < n, and prime(k)^3 + 2*prime(n-k)^3 is prime for some 0 < k < n.
(iii) If n > 5, then prime(k)^3 + 2*p(n-k)^3 is prime for some 0 < k < n, where p(.) is the partition function (A000041). If n > 2, then prime(k)^3 + 2*q(n-k)^3 is prime for some 0 < k < n, where q(.) is the strict partition function (A000009).

			a(4) = 1 since prime(3)^2 + 4*prime(1)^2 = 5^2 + 4*2^2 = 41 is prime.
a(6) = 1 since prime(5)^2 + 4*prime(1)^2 = 11^2 + 4*2^2 = 137 is prime.
a(8) = 1 since prime(3)^2 + 4*prime(5)^2 = 5^2 + 4*11^2 = 509 is prime.
a(16) = 1 since prime(6)^2 + 4*prime(10)^2 = 13^2 + 4*29^2 = 3533 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A002313, A220413, A232174, A232269, A232465, A232502, A233150, A233204, A233206, A233296, A233307, A233346.

a[n_]:=Sum[If[PrimeQ[Prime[k]^2+4*Prime[n-k]^2],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A233529 a(n) = |{0 < k <= n/2: prime(k)*prime(n-k) - 6 is prime}|.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 1, 4, 5, 1, 5, 3, 2, 1, 2, 5, 5, 4, 5, 6, 5, 5, 4, 8, 5, 7, 4, 3, 6, 6, 4, 8, 6, 7, 7, 8, 7, 5, 5, 5, 7, 8, 6, 13, 9, 5, 3, 9, 6, 8, 11, 5, 9, 9, 10, 8, 9, 14, 9, 10, 13, 11, 6, 9, 12, 10, 12, 14, 10, 12, 7, 13, 9, 7, 7, 15, 12, 6, 10, 11, 12, 12, 9, 18, 15, 14, 11, 10, 10, 8, 13, 21, 9, 14

Offset: 1

Zhi-Wei Sun, Dec 11 2013

nonn

Conjectures:
(i) a(n) > 0 for all n > 5. Also, for any n > 5, 2*prime(k)*prime(n-k) - 3 is prime for some 0 < k < n.
(ii) For any n > 1 not among 3, 9, 13, 26, there is a positive integer k < n with prime(k)*prime(n-k) - 2 prime.  For any n > 2 not among 8, 23, 33, there is a positive integer k < n with prime(k)*prime(n-k) - 4 prime.

			a(8) = 1 since prime(4)*prime(4) - 6 = 7*7 - 6 = 43 is prime.
a(10) = 1 since prime(3)*prime(7) - 6 = 5*17 - 6 = 79 is prime.
a(16) = 1 since prime(3)*prime(13) - 6 = 5*41 - 6 = 199 is prime.
a(20) = 1 since prime(7)*prime(13) - 6 = 17*41 - 6 = 691 is 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, A232465, A232502, A232861, A233150, A233204, A233206, A233439.

PQ[n_]:=n>0&&PrimeQ[n]
a[n_]:=Sum[If[PQ[Prime[k]*Prime[n-k]-6],1,0],{k,1,n/2}]
Table[a[n],{n,1,100}]

A238585 Number of primes p < n with prime(p)^2 + (prime(n)-1)^2 prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 01 2014

nonn

Conjecture: (i) a(n) > 0 unless n divides 6, and a(n) = 1 only for n = 4, 5, 7, 10, 11, 12, 19, 21, 22, 31, 42, 44.
(ii) If n > 2 is not equal to 9, then prime(n)^2 + (prime(p) - 1)^2 is prime for some prime p < n.
(iii) For n > 3, there is a prime p < n with prime(p) + prime(n) + 1 prime. If n > 9 is not equal to 18, then prime(p)^2 + prime(n)^2 - 1 is prime for some prime p < n.

			a(7) = 1 since 3 and prime(3)^2 + (prime(7)-1)^2 = 5^2 + 16^2 = 281 are both prime.
a(44) = 1 since 23 and prime(23)^2 + (prime(44)-1)^2 = 83^2 + 192^2 = 43753 are 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, A232465, A238580.

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

cp's OEIS Frontend

A237496 Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m) - 2 prime, where pi(.) is given by A000720.

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A232502 Number of ways to write n = k + m (0 < k < m) with 2*prime(m) - prime(k) prime.

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A233296 a(n) = |{0 < k < n: k*prime(n-k) + 1 is prime}|.

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A233439 a(n) = |{0 < k < n: prime(k)^2 + 4*prime(n-k)^2 is prime}|.

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A233529 a(n) = |{0 < k <= n/2: prime(k)*prime(n-k) - 6 is prime}|.

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A238585 Number of primes p < n with prime(p)^2 + (prime(n)-1)^2 prime.

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