A236256 -id:A236256 - 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}]

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}]

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}]

A236263 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 21 2014

nonn

It seems that a(n) > 0 for all n > 17. (We have verified this for n up to 13000.) If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! + prime(m) prime.

See also A236265 for a similar sequence.

			a(18) = 1 since phi(3)/2 + phi(15)/8 = 1 + 1 = 2 with 2! + prime(2) = 2 + 3 = 5 prime.
a(356) = 1 since phi(203)/2 + phi(153)/8 = 84 + 12 = 96 with 96! + prime(96) = 96! + 503 prime.
a(457) = 1 since phi(7)/2 + phi(450)/8 = 3 + 15 = 18 with 18! + prime(18) = 18! + 61 = 6402373705728061 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..8100

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236265.

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

A236265 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! - prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 21 2014

nonn

It seems that a(n) > 0 for all n > 21. If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! - prime(m) prime.

See also A236263 for a similar sequence.

			a(23) = 1 since phi(7)/2 + phi(16)/8 = 3 + 1 = 4 with 4! - prime(4) = 24 - 7 = 17 prime.
a(26) = 1 since phi(9)/2 + phi(17)/8 = 3 + 2 = 5 with 5! - prime(5) = 120 - 11 = 109 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263.

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

A236325 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 22 2014

nonn

It might seem that a(n) > 0 for all n > 14, but a(7365) = 0. If a(n) > 0 infinitely often, then there are infinitely many positive integers m with m! + prime(m) prime.

			a(10) = 1 since phi(1)/2 + phi(9)/12 = 1/2 + 6/12 = 1 with 1! + prime(1) = 1 + 2 = 3 prime.
a(23) = 1 since phi(10)/2 + phi(13)/12 = 2 + 1 = 3 with 3! + prime(3) = 6 + 5 = 11 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

Cf. A000010, A000040, A000142, A063499, A064278, A064401, A236241, A236256, A236263, A236265.

p[n_]:=IntegerQ[n]&&PrimeQ[n!+Prime[n]]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236344 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2^m + prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 22 2014

nonn

a(n) = 0 for n = 1, ..., 15, 481, 564, 66641, 70965, 72631, .... If a(n) > 0 infinitely often, then there are infinitely many positive integers m with 2^m + prime(m) prime.

			a(26) = 1 since phi(5)/2 + phi(21)/12 = 2 + 1 = 3 with 2^3 + prime(3) = 8 + 5 = 13 prime.
a(5907) = 1 since phi(3944)/2 + phi(5907-3944)/12 = 896 + 150 = 1046 with 2^(1046) + prime(1046) = 2^(1046) + 8353 prime.

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

Cf. A000010, A000040, A000079, A077375, A236241, A236256.

p[n_]:=IntegerQ[n]&&PrimeQ[2^n+Prime[n]]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

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

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A236263 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

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A236265 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with m! - prime(m) prime}|, where phi(.) is Euler's totient function.

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A236325 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with m! + prime(m) prime}|, where phi(.) is Euler's totient function.

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A236344 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2^m + prime(m) prime}|, where phi(.) is Euler's totient function.

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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.

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.

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