A236568 -id:A236568 - cp's OEIS frontend

A236569 Least term p of A236568 with 2*n - p prime, or 0 if such a prime p does not exist.

0, 0, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 11, 3, 3, 5, 31, 3, 5, 3, 3, 5, 3, 5, 11, 3, 5, 31, 3, 3, 5, 31, 3, 5, 3, 3, 5, 43, 3, 5, 3, 5, 11, 3, 5, 43, 31, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 11, 43, 11, 43, 31, 3, 5, 3, 5, 11, 3

Offset: 1

Zhi-Wei Sun, Jan 29 2014

nonn

The conjecture in A236566 implies that a(n) > 0 for all n > 2.

			a(3) = 3 since prime(3 + 2) + 2 = 11 + 2 = 13 and 2*3 - 3 = 3 are both prime, but prime(2 + 2) + 2 = 9 is not.

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

Cf. A000040, A020481, A236566, A236568.

p[m_]:=PrimeQ[Prime[m+2]+2]
Do[Do[If[p[Prime[k]]&&PrimeQ[2n-Prime[k]],Print[n," ",Prime[k]];Goto[aa]],{k,1,PrimePi[2n-1]}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

A237283 Primes p with prime(prime(p)) + 2 also prime.

Original entry on oeis.org

2, 3, 7, 13, 23, 29, 59, 71, 103, 193, 257, 271, 281, 311, 317, 389, 433, 439, 463, 569, 577, 619, 673, 683, 691, 797, 811, 857, 859, 887, 1031, 1069, 1109, 1129, 1153, 1229, 1307, 1597, 1613, 1867, 1949, 1951, 2069, 2297, 2477, 2551, 2621, 2657, 2699, 2753

Offset: 1

Zhi-Wei Sun, Feb 05 2014

nonn

This sequence is interesting because of the conjecture in A237253.

A236481, A236482 and A236484 are subsequences of the sequence.

			a(1) = 2 since 2 and prime(prime(2)) + 2 = prime(3) + 2 = 7 are both prime.

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

Cf. A000040, A001359, A006512, A096478, A218829, A236457, A236458, A236467, A236481, A236482, A236484, A236568, A237253.

n=0;Do[If[PrimeQ[Prime[Prime[Prime[k]]]+2],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
Select[Prime[Range[500]],PrimeQ[Prime[Prime[#]]+2]&] (* Harvey P. Dale, May 30 2018 *)

A236573 Number of ordered ways to write n = k + m (k > 0, m > 0) such that p = 2k + phi(m) - 1, prime(p + 2) + 2 and 2n - p are all prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 29 2014

nonn

Conjecture: a(n) > 0 for all n > 712.

This implies the conjecture in A236566.

			a(100) = 1 since 100 = 10 + 90 with 2*10 + phi(90) - 1 = 20 + 24 - 1 = 43, prime(43 + 2) + 2 = 197 + 2 = 199 and 2*100 - 43 = 157 all prime.
a(1727) = 1 since 1727 = 956 + 771 with 2*956 + phi(771) - 1 = 1912 + 512 - 1 = 2423, prime(2423 + 2) + 2 = 21599 + 2 = 21601 and 2*1727 - 2423 = 1031 all prime.

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

Cf. A000010, A000040, A001359, A002372, A002375, A006512, A234808, A236531, A236566, A236568, A236569.

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

cp's OEIS Frontend

A236569 Least term p of A236568 with 2*n - p prime, or 0 if such a prime p does not exist.

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A237283 Primes p with prime(prime(p)) + 2 also prime.

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A236573 Number of ordered ways to write n = k + m (k > 0, m > 0) such that p = 2k + phi(m) - 1, prime(p + 2) + 2 and 2n - p are all prime, where phi(.) is Euler's totient function.

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cp's OEIS Frontend

A236569 Least term p of A236568 with 2*n - p prime, or 0 if such a prime p does not exist.

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Mathematica

A237283 Primes p with prime(prime(p)) + 2 also prime.

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Author

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Mathematica

A236573 Number of ordered ways to write n = k + m (k > 0, m > 0) such that p = 2*k + phi(m) - 1, prime(p + 2) + 2 and 2*n - p are all prime, where phi(.) is Euler's totient function.

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Mathematica

A236573 Number of ordered ways to write n = k + m (k > 0, m > 0) such that p = 2k + phi(m) - 1, prime(p + 2) + 2 and 2n - p are all prime, where phi(.) is Euler's totient function.