A235681 -id:A235681 - cp's OEIS frontend

A235682 Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function.

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

Offset: 1

Zhi-Wei Sun, Jan 13 2014

nonn

Conjecture: a(n) > 0 for all n > 84.

Clearly, this implies that there are infinitely many primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.

			a(10) = 1 since 10 = 1 + 9 with phi(1) + phi(9)/2 + 1 = 5, prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 all prime.
a(95) = 1 since 95 = 62 + 33 with phi(62) + phi(33)/2 + 1 = 41, prime(41) - 41 + 1 = 139 and 41*42 - prime(41) = 1543 all prime.
a(421) = 1 since 421 = 289 + 132 with phi(289) + phi(132)/2 + 1 = 293, prime(293) - 293 + 1 = 1621 and 293*294 - prime(293) = 84229 all prime.

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

Cf. A000010, A000040, A232353, A234694, A234695, A235189, A235330, A235613, A235614, A235661, A235681.

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

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

Original entry on oeis.org

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}]

A235508 Number of ways to write 2n = p + q with q > 0 such that p, p(p+1) - prime(p) and prime(q) - q + 1 are all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 14 2014

nonn

Conjecture: a(n) > 0 for all n > 1.

			a(7) = 1 since 2*7 = 11 + 3 with 11, 11*12 - prime(11) = 101 and prime(3) - 3 + 1 = 3 all prime.
a(19) = 1 since 2*19 = 37 + 1 with 37, 37*38 - prime(37) = 1249 and prime(1) - 1 + 1 = 2 all prime.
a(98) = 1 since 2*98 = 11 + 185 with 11, 11*12 - prime(11) = 101 and prime(185) - 185 + 1 = 919 all prime.

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

Cf. A000040, A234695, A234851, A235189, A235330, A235661, A235681.

p[k_]:=PrimeQ[Prime[k](Prime[k]+1)-Prime[Prime[k]]]
q[m_]:=PrimeQ[Prime[m]-m+1]
a[n_]:=Sum[If[p[k]&&q[2n-Prime[k]],1,0],{k,1,PrimePi[2n-1]}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A235682 Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function.

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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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A235508 Number of ways to write 2n = p + q with q > 0 such that p, p(p+1) - prime(p) and prime(q) - q + 1 are all prime.

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cp's OEIS Frontend

A235682 Number of ways to write n = k + m with k > 0 and m > 2 such that p = phi(k) + phi(m)/2 + 1, prime(p) - p + 1 and p*(p+1) - prime(p) are all prime, where phi(.) is Euler's totient function.

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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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A235508 Number of ways to write 2*n = p + q with q > 0 such that p, p*(p+1) - prime(p) and prime(q) - q + 1 are all prime.

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A235508 Number of ways to write 2n = p + q with q > 0 such that p, p(p+1) - prime(p) and prime(q) - q + 1 are all prime.