A085637 -id:A085637 - cp's OEIS frontend

A232442 a(n) = |{0 < k < n: mprime(m) - 1 and mprime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 6, 1, 2, 2, 0, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 4, 1, 1, 0, 1, 2, 2, 2, 4, 0, 0, 1, 2, 0, 3, 3, 3, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1, 14, 3, 2, 2, 2, 2, 3, 4, 5, 3, 2, 3, 1, 3, 3, 4, 6, 3, 0, 5, 3, 1, 0, 5, 2, 0, 3, 6, 1

Offset: 1

Zhi-Wei Sun, Jan 14 2014

nonn

Conjecture: a(n) > 0 for all n > 214.
This implies that there are infinitely many twin prime pairs of the special form {m*prime(m) - 1, m*prime(m) + 1}.
We have verified the conjecture for n up to 10^5.

			a(25) = 1 since sigma(6) + phi(19) = 12 + 18 = 30 with {30*prime(30) - 1, 30*prime(30) + 1} = {3389, 3391} a twin prime pair.
a(100) = 1 since sigma(75) + phi(25) = 124 + 20 = 144 with {144*prime(144) - 1, 144*prime(144) + 1} = {119087, 119089} a twin prime pair.

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

Cf. A000010, A000040, A000203, A085637, A086172, A086173, A232270, A232861, A233544, A233539.

sigma[n_]:=DivisorSigma[1,n]
q[n_]:=PrimeQ[n*Prime[n]-1]&&PrimeQ[n*Prime[n]+1]
f[n_,k_]:=sigma[k]+EulerPhi[n-k]
a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A257789 Numbers n such that 2nprime(n) - 1 and 2nprime(n) + 1 are both prime.

Original entry on oeis.org

1, 2, 3, 24, 30, 33, 54, 90, 156, 168, 189, 225, 294, 300, 402, 576, 741, 780, 825, 849, 918, 948, 978, 1014, 1245, 1542, 1551, 1608, 1614, 1617, 1770, 1773, 1908, 1914, 1920, 1947, 2025, 2286, 2361, 2370, 2598, 2760, 2865, 2970, 3081, 3516, 3744, 3759, 3948, 4023

Offset: 1

Juri-Stepan Gerasimov, May 08 2015

a(n) is divisible by 3 for n >= 3. - Robert Israel, May 08 2015

			2 is in this sequence because 2*2*prime(2) - 1 = 11 and 2*2*prime(2) + 1 = 13 are both prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A085637.

[n: n in [1..4500] | IsPrime(2*n*NthPrime(n)-1) and IsPrime(2*n*NthPrime(n)+1)];

filter:= proc(n)
local p;
p:= ithprime(n);
isprime(2*n*p+1) and isprime(2*n*p-1)
end proc:
select(filter, [1,2,seq(3*j,j=1..10^5)]); # Robert Israel, May 08 2015

Select[Range[3000], PrimeQ[2 # Prime[#] - 1] && PrimeQ[2 #  Prime[#] + 1] &] (* Vincenzo Librandi, May 09 2015 *)
Select[Range[4200],AllTrue[2# Prime[#]+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 08 2018 *)

v=List(); n=0; forprime(p=2,1e5, n++; if(isprime(2*n*p-1) && isprime(2*n*p+1), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, May 08 2015

cp's OEIS Frontend

A232442 a(n) = |{0 < k < n: mprime(m) - 1 and mprime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A257789 Numbers n such that 2nprime(n) - 1 and 2nprime(n) + 1 are both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

cp's OEIS Frontend

A232442 a(n) = |{0 < k < n: m*prime(m) - 1 and m*prime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A257789 Numbers n such that 2n*prime(n) - 1 and 2n*prime(n) + 1 are both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A232442 a(n) = |{0 < k < n: mprime(m) - 1 and mprime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

A257789 Numbers n such that 2nprime(n) - 1 and 2nprime(n) + 1 are both prime.