A235806 -id:A235806 - cp's OEIS frontend

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.

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}]

A235920 Primes p with prime(p) - p + 1 and (p^2 - 1)/4 - prime(p) both prime.

Original entry on oeis.org

17, 31, 41, 43, 61, 71, 83, 103, 109, 173, 181, 211, 271, 349, 353, 541, 661, 673, 743, 811, 911, 953, 971, 1171, 1429, 1471, 1483, 1723, 1787, 2053, 2203, 2579, 2749, 3019, 3299, 3391, 3433, 3463, 3727, 3917, 4003, 4021, 4049, 4243, 4447, 4567, 4657, 4729, 4801, 4993

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

By the conjecture in A235919, this sequence should have infinitely many terms.

			a(1) = 17 with prime(17) - 17 + 1 =   59 - 16 = 43 and (17^2 - 1)/4 - prime(17) = 72 - 59 = 13 both prime.

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

Cf. A000040, A234695, A235727, A235806, A235914, A235919.

PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PrimeQ[Prime[n]-n+1]&&PQ[(n^2-1)/4-Prime[n]]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]

cp's OEIS Frontend

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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A235920 Primes p with prime(p) - p + 1 and (p^2 - 1)/4 - prime(p) both prime.

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cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A235920 Primes p with prime(p) - p + 1 and (p^2 - 1)/4 - prime(p) both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.