A234808 -id:A234808 - cp's OEIS frontend

A234809 a(n) = |{0 < k < n: p = k + phi(n-k) and 2*(n-p) + 1 are both prime}|, where phi(.) is Euler's totient function.

0, 0, 1, 2, 1, 3, 1, 4, 1, 1, 1, 5, 3, 7, 3, 1, 1, 7, 5, 9, 4, 2, 1, 9, 5, 2, 4, 3, 1, 10, 5, 14, 2, 2, 2, 1, 6, 14, 5, 4, 1, 15, 5, 16, 5, 5, 3, 17, 8, 4, 5, 6, 3, 17, 7, 5, 2, 6, 6, 17, 11, 25, 3, 5, 3, 1, 11, 25, 4, 4, 4, 22, 10, 26, 6, 7, 8, 3, 9, 26, 7, 9, 6, 25, 8, 3, 7, 9, 10, 25, 15, 6, 2, 9, 9, 2, 13, 29, 3, 7

Offset: 1

Zhi-Wei Sun, Dec 30 2013

nonn

Conjecture: a(n) > 0 for all n > 2.
Clearly, this implies Lemoine's conjecture which states that any odd number 2*n + 1 > 5 can be written as 2*p + q with p and q both prime.
See also A234808 for a similar conjecture.

			a(5) = 1 since 1 + phi(4) = 3 and 2*(5-3) + 1 = 5 are both prime.
a(16) = 1 since 7 + phi(9) = 13 and 2*(16-13) + 1 = 7 are both prime.
a(41) = 1 since 7 +phi(34) = 23 and 2*(41-23) + 1 = 37 are both prime.
a(156) = 1 since 131 + phi(25) = 151 and 2*(156-151) + 1 = 11 are both prime.

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

Cf. A000010, A000040, A046927, A234470, A234475, A234514, A234567, A234615, A234694, A234808

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

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

A234809 a(n) = |{0 < k < n: p = k + phi(n-k) and 2*(n-p) + 1 are both prime}|, where phi(.) is Euler's totient function.

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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

A234809 a(n) = |{0 < k < n: p = k + phi(n-k) and 2*(n-p) + 1 are both 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 = 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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Examples

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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.