A235728 -id:A235728 - cp's OEIS frontend

A235727 Odd primes p with (p^2 - 1)/4 - prime((p - 1)/2) and (p^2 - 1)/4 + prime((p - 1)/2) both prime.

7, 11, 19, 23, 41, 73, 83, 109, 197, 211, 229, 271, 379, 461, 541, 631, 641, 659, 859, 991, 1031, 1049, 1051, 1093, 1103, 1217, 1429, 1451, 1879, 2063, 2131, 2287, 2341, 2411, 3019, 3257, 3461, 3659, 3673, 3691, 3709, 3917, 3967, 4409, 4463, 4519, 5279, 5303, 5471, 5477

Offset: 1

Zhi-Wei Sun, Jan 15 2014

nonn

By the conjecture in A235728, this sequence should have infinitely many terms.

			a(1) = 7 since neither (3^2-1)/4 - prime((3-1)/2) = 1 nor (5^2-1)/4 + prime((5-1)/2) = 9 is prime, but (7^2-1)/4 - prime((7-1)/2) = 12 - 5 = 7 and (7^2-1)/4 + prime((7-1)/2) = 12 + 5 = 17 are both prime.

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

Cf. A000040, A232353, A235592, A235661, A235681, A235682, A235703, A235728.

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

A235912 a(n) = |{0 < k < n - 2: 2m + 1, m(m-1) - prime(m) and m*(m+1) - prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 2, 4, 2, 6, 5, 6, 7, 4, 8, 7, 8, 8, 11, 7, 12, 9, 9, 12, 5, 14, 10, 9, 9, 9, 9, 7, 8, 11, 9, 8, 7, 14, 8, 6, 9, 5, 5, 9, 11, 3, 9, 6, 13, 8, 8, 6, 7, 6, 5, 4, 3, 1, 5, 5, 5, 6, 5, 7, 7, 4, 7, 11, 8, 3, 5, 3, 10, 4, 4, 3, 9, 2, 4, 4, 5, 8, 12, 13, 4, 9, 5, 11, 5, 12, 7, 4, 4

Offset: 1

Zhi-Wei Sun, Jan 16 2014

nonn

Conjecture: a(n) > 0 for all n > 11.

This implies that there are infinitely many odd primes p = 2*m + 1 with q = m*(m-1) - prime(m) and r = m*(m+1) - prime(m) both prime. Note that r - q = 2*m.

			 a(10) = 1 since phi(5) + phi(5)/2 = 6 with 2*6 + 1 = 13, 5*6 - prime(6) = 30 - 13 = 17 and 6*7 - prime(6) = 42 - 13 = 29 all prime.

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

Cf. A000010, A000040, A235592, A235728.

PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PrimeQ[2n+1]&&PQ[n(n-1)-Prime[n]]&&PQ[n(n+1)-Prime[n]]
f[n_,k_]:=EulerPhi[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}]

A235805 a(n) = |{0 < k < n - 2: 2m + 1, m(m+1) - prime(m) and m*(m+1) - prime(m+1) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 2, 4, 4, 4, 4, 4, 4, 2, 6, 3, 7, 4, 2, 7, 3, 5, 4, 4, 6, 6, 6, 4, 4, 7, 8, 9, 6, 6, 11, 8, 10, 6, 6, 12, 8, 13, 6, 12, 8, 13, 10, 7, 14, 10, 11, 11, 11, 16, 14, 13, 9, 15, 11, 23, 14, 12, 11, 12, 10, 14, 8, 15, 9, 14, 13, 11, 12, 9, 19, 9, 14, 11, 16, 8, 14, 5, 13, 8, 13, 9, 13, 10, 15, 10, 11, 12, 17, 8, 13, 10, 11, 7, 18

Offset: 1

Zhi-Wei Sun, Jan 16 2014

nonn

Conjecture: a(n) > 0 for all n > 5.

This implies that there are infinitely many odd primes p = 2*m + 1 with m*(m+1) - prime(m) and m*(m+1)- prime(m+1) both prime.

			a(8) = 2 since phi(4) + phi(4)/2 = 3 with 2*3 + 1 = 7, 3*4 - prime(3) = 7 and 3*4 - prime(4) = 5 all prime, and phi(5) + phi(3)/2 = 5 with 2*5 + 1 = 11, 5*6 - prime(5) = 19 and 5*6 - prime(6) = 17 all prime.

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

Cf. A000010, A000040, A235592, A235728, A235806.

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

cp's OEIS Frontend

A235727 Odd primes p with (p^2 - 1)/4 - prime((p - 1)/2) and (p^2 - 1)/4 + prime((p - 1)/2) both prime.

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A235912 a(n) = |{0 < k < n - 2: 2m + 1, m(m-1) - prime(m) and m*(m+1) - prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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A235805 a(n) = |{0 < k < n - 2: 2m + 1, m(m+1) - prime(m) and m*(m+1) - prime(m+1) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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cp's OEIS Frontend

A235727 Odd primes p with (p^2 - 1)/4 - prime((p - 1)/2) and (p^2 - 1)/4 + prime((p - 1)/2) both prime.

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Mathematica

A235912 a(n) = |{0 < k < n - 2: 2*m + 1, m*(m-1) - prime(m) and m*(m+1) - prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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Mathematica

A235805 a(n) = |{0 < k < n - 2: 2*m + 1, m*(m+1) - prime(m) and m*(m+1) - prime(m+1) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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Mathematica

A235912 a(n) = |{0 < k < n - 2: 2m + 1, m(m-1) - prime(m) and m*(m+1) - prime(m) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

A235805 a(n) = |{0 < k < n - 2: 2m + 1, m(m+1) - prime(m) and m*(m+1) - prime(m+1) are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.