A236249 -id:A236249 - cp's OEIS frontend

A236241 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/8 is an integer with C(2m, m) + prime(m) prime}|, where C(2m, m) = (2*m)!/(m!)^2, and phi(.) is Euler's totient function.

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

Offset: 1

Zhi-Wei Sun, Jan 20 2014

nonn

Conjecture: a(n) > 0 for every n = 20, 21, ... .
We have verified this for n up to 75000.
The conjecture implies that there are infinitely many primes of the form C(2*m, m) + prime(m).
See A236245 for primes of the form C(2*m, m) + prime(m). See also A236242 for a list of known numbers m with C(2*m, m) + prime(m) prime.

			a(20) = 1 since phi(5) + phi(15)/8 = 4 + 1 = 5 with C(2*5,5) + prime(5) = 252 + 11 = 263 prime.
a(330) = 1 since phi(211) + phi(330-211)/8 = 210 + 96/8 = 222 with C(2*222,222) + prime(222) = C(444,222) + 1399 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. A000010, A000040, A000984, A236242, A236245, A236248, A236249, A236256.

p[n_]:=IntegerQ[n]&&PrimeQ[Binomial[2n,n]+Prime[n]]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/8
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236256 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/4 is an integer with C(2m, m) - prime(m) prime}|, where C(2m, m) = (2*m)!/(m!)^2.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 21 2014

nonn

Conjecture: a(n) > 0 for all n > 410.
This implies that there are infinitely many positive integers m with C(2*m, m) - prime(m) prime. We have verified the conjecture for n up to 51000.
See A236248 for a list of known numbers m with C(2*m, m) - prime(m) prime.
See also A236249 for those primes of the form C(2*m, m) - prime(m).

			a(12) = 1 since phi(2) + phi(10)/4 = 1 + 1 = 2 with C(2*2, 2) - prime(2) = 6 - 3 = 3 prime.
a(33) = 1 since phi(1) + phi(32)/4 = 1 + 4 = 5 with C(2*5, 5) - prime(5) = 252 - 11 = 241 prime.
a(697) = 1 since phi(452) + phi(697-452)/4 = 224 + 42 = 266 with C(2*266, 266) - prime(266) = C(532, 266) - 1699 prime.

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

Cf. A000010, A000040, A000984, A236241, A236242, A236245, A236248, A236249.

p[n_]:=IntegerQ[n]&&PrimeQ[Binomial[2n,n]-Prime[n]]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236242 Numbers m with C(2m, m) + prime(m) prime, where C(2m, m) = (2*m)!/(m!)^2.

Original entry on oeis.org

5, 6, 7, 8, 12, 13, 19, 69, 91, 102, 116, 119, 171, 198, 216, 222, 278, 299, 338, 584, 722, 774, 874, 978, 1004, 1163, 1268, 1492, 1836, 1932, 1966, 2982, 3508, 3964, 4264, 4894, 5028, 8236, 8552, 8639, 12749, 14017, 14402, 18150, 18321, 18514, 18979, 20935, 21815, 21828, 21890, 30734

Offset: 1

Zhi-Wei Sun, Jan 20 2014

nonn

According to the conjecture in A236241, this sequence should have infinitely many terms. The prime C(2*a(52),a(52)) + prime(a(52)) = C(61468, 30734) + prime(30734) has 18502 decimal digits.
For primes of the form C(2*m, m) + prime(m), see A236245.
See also A236248 for a similar sequence.

			a(1) = 5 since C(2*1,1) + prime(1) = 4, C(2*2,2) + prime(2) = 9, C(2*3,3) + prime(3) = 25 and C(2*4,4) + prime(4) = 77 are all composite, but C(2*5,5) + prime(5) = 252 + 11 = 263 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..52
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A000984, A236241, A236245, A236248, A236249, A236256.

n=0;Do[If[PrimeQ[Binomial[2m,m]+Prime[m]],n=n+1;Print[n," ",m]],{m,1,10000}]
Select[Range[9000],PrimeQ[Binomial[2#,#]+Prime[#]]&] (* Harvey P. Dale, Jan 18 2016 *)

a(41)-a(52) from bfile by Robert Price, Aug 31 2019

A236248 Numbers m with C(2m, m) - prime(m) prime, where C(2m, m) = (2*m)!/(m!)^2.

Original entry on oeis.org

2, 5, 6, 10, 29, 132, 266, 322, 350, 538, 667, 693, 776, 977, 1336, 1810, 1908, 1980, 2175, 2616, 2716, 3211, 3473, 5223, 5630, 5758, 6585, 6979, 7964, 8469, 9052, 9758, 10324, 16876, 25760, 28171

Offset: 1

Zhi-Wei Sun, Jan 21 2014

nonn

According to the conjecture in A236256, this sequence should have infinitely many terms.
The prime C(2*a(36), a(36)) - prime(a(36)) = C(56342, 28171) - prime(28171) has 16959 decimal digits.
See A236249 for primes of the form C(2*m, m) - prime(m).
See also A236242 for a similar sequence.

			a(1) = 2 since C(2*1, 1) - prime(1) = 2 - 2 = 0 is not prime, but C(2*2, 2) - prime(2) = 6 - 3 = 3 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..36

Cf. A000040, A000984, A236241, A236242, A236245, A236249, A236256.

n=0;Do[If[PrimeQ[Binomial[2m,m]-Prime[m]],n=n+1;Print[n," ",m]],{m,1,10000}]

cp's OEIS Frontend

A236241 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/8 is an integer with C(2*m, m) + prime(m) prime}|, where C(2*m, m) = (2*m)!/(m!)^2, and phi(.) is Euler's totient function.

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A236256 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/4 is an integer with C(2*m, m) - prime(m) prime}|, where C(2*m, m) = (2*m)!/(m!)^2.

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A236242 Numbers m with C(2*m, m) + prime(m) prime, where C(2*m, m) = (2*m)!/(m!)^2.

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A236248 Numbers m with C(2*m, m) - prime(m) prime, where C(2*m, m) = (2*m)!/(m!)^2.

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A236241 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/8 is an integer with C(2m, m) + prime(m) prime}|, where C(2m, m) = (2*m)!/(m!)^2, and phi(.) is Euler's totient function.

A236256 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/4 is an integer with C(2m, m) - prime(m) prime}|, where C(2m, m) = (2*m)!/(m!)^2.

A236242 Numbers m with C(2m, m) + prime(m) prime, where C(2m, m) = (2*m)!/(m!)^2.

A236248 Numbers m with C(2m, m) - prime(m) prime, where C(2m, m) = (2*m)!/(m!)^2.