A236242 -id:A236242 - 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}]

A236245 Primes of the form C(2m, m) + prime(m), where C(2m, m) = (2*m)!/(m!)^2.

Original entry on oeis.org

263, 937, 3449, 12889, 2704193, 10400641, 35345263867, 23623985175715118288974865541854103729347, 362048725489728431058442528694228154899210914562190067

Offset: 1

Zhi-Wei Sun, Jan 20 2014

nonn

Although the primes in this sequence are very rare, by the conjecture in A236241 there should be infinitely many such primes.

See A236242 for a list of known numbers m with C(2*m, m) + prime(m) prime.

			a(1) = 263 since C(2*5, 5) + prime(5) = 252 + 11 = 263 is prime, and those C(2*m, m) + prime(m) with 0 < m < 5 are composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..25

Cf. A000040, A000984, A236241, A236242.

s[n_]:=Binomial[2n,n]+Prime[n]
a[n_]:=s[A236242(n)]
Table[a[n],{n,1,40}]

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}]

A236249 Primes of the form C(2m, m) - prime(m), where C(2m, m) = (2*m)!/(m!)^2.

Original entry on oeis.org

3, 241, 911, 184727, 30067266499540931, 1454272161238683681127450712107767894181359647011258114106151524833563647084221

Offset: 1

Zhi-Wei Sun, Jan 21 2014

nonn

Though the primes in this sequence are very rare, according to the conjecture in A236256 there should be infinitely many such primes.
See A236248 for a list of known numbers m with C(2*m, m) - prime(m) prime.
See also A236245 for a similar sequence.

			a(1) = 3 since C(2*1, 1) - prime(1) = 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..12

Cf. A000040, A000984, A236241, A236242, A236245, A236248, A236256.

t[n_]:=Binomial[2n,n]-Prime[n]
a[n_]:=t[A234248(n)]
Table[a[n],{n,1,6}]

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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A236245 Primes of the form C(2*m, m) + prime(m), 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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Mathematica

A236249 Primes of the form C(2*m, m) - prime(m), where C(2*m, m) = (2*m)!/(m!)^2.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

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.

A236245 Primes of the form C(2m, m) + prime(m), 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.

A236249 Primes of the form C(2m, m) - prime(m), where C(2m, m) = (2*m)!/(m!)^2.