A235912 -id:A235912 - cp's OEIS frontend

A235914 Odd primes p = 2m + 1 with m(m-1) - prime(m) and m*(m+1) - prime(m) both prime.

13, 17, 23, 29, 31, 43, 73, 89, 181, 229, 313, 367, 379, 557, 631, 683, 1021, 1069, 1093, 1151, 1303, 1459, 1471, 1663, 1733, 1831, 1871, 2411, 2473, 2791, 2843, 2887, 3673, 3691, 3793, 3797, 3863, 4001, 4139, 4261, 5261, 5431, 6091, 6301, 6661, 6737, 6883, 7489, 7523, 7873

Offset: 1

Zhi-Wei Sun, Jan 16 2014

nonn

By the conjecture in A235912, this sequence should have infinitely many terms.

			a(1) = 13 since none of 1*2 - prime(1) = 0, 1*2 - prime(2) = -1, 2*3 - prime(3) = 1 and 2*4 + 1 = 9 = 4*5 - prime(5) is prime, but 2*6 + 1 = 13, 5*6 - prime(6) = 30 - 13 = 17 and 6*7 - prime(6) = 42 - 13 = 29 are all prime.

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

Cf. A000040, A235727, A235912.

PQ[n_]:=n>0&&PrimeQ[n]
q[n_]:=PQ[n(n-1)-Prime[n]]&&PQ[n(n+1)-Prime[n]]
n=0;Do[If[q[(Prime[k]-1)/2],n=n+1;Print[n," ",Prime[k]]],{k,2,1000}]

A235917 a(n) = |{0 < k < n - 2: p = prime(k) + phi(n-k)/2, p^2 - 1 - prime(p) and (p^2 - 1)/2 - prime(p) are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 14.
(ii) For any integer n > 12, there is a positive integer k < n such that p = prime(k) + phi(n-k), (p^2 - 1)/2 - prime(p) and (p^2 - 1)/4 - prime(p) are all prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with p^2 - 1 - prime(p) and (p^2 - 1)/2 - prime(p) both prime.

			a(10) = 1 since prime(5) + phi(5)/2 = 11 + 2 = 13, 13^2 - 1 - prime(13) = 168 - 41 = 127 and (13^2 - 1)/2 - prime(13) = 84 - 41 = 43 are all prime.
a(71) = 1 since prime(19) + phi(52)/2 = 67 + 12 = 79, 79^2 - 1 - prime(79) = 6240 - 401 = 5839 and (79^2 - 1)/2 - prime(79) = 3120 - 401 = 2719 are all prime.

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

Cf. A000010, A000040, A234694, A235912.

PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PrimeQ[n]&&PQ[n^2-1-Prime[n]]&&PQ[(n^2-1)/2-Prime[n]]
f[n_,k_]:=Prime[k]+EulerPhi[n-k]/2
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A235914 Odd primes p = 2m + 1 with m(m-1) - prime(m) and m*(m+1) - prime(m) both prime.

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A235917 a(n) = |{0 < k < n - 2: p = prime(k) + phi(n-k)/2, p^2 - 1 - prime(p) and (p^2 - 1)/2 - prime(p) are all prime}|, where phi(.) is Euler's totient function.

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cp's OEIS Frontend

A235914 Odd primes p = 2*m + 1 with m*(m-1) - prime(m) and m*(m+1) - prime(m) both prime.

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A235917 a(n) = |{0 < k < n - 2: p = prime(k) + phi(n-k)/2, p^2 - 1 - prime(p) and (p^2 - 1)/2 - prime(p) are all prime}|, where phi(.) is Euler's totient function.

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A235914 Odd primes p = 2m + 1 with m(m-1) - prime(m) and m*(m+1) - prime(m) both prime.