A238506 Number of primes of the form: |P_i +- P_n / P_i| or |P_i/2 +- 2P_n / P_i|, where 2 <= i <= n, P_i = the i-th primorial number (A002110).

3, 7, 10, 11, 16, 13, 13, 15, 10, 12, 14, 16, 11, 13, 7, 21, 14, 25, 13, 10, 12, 9, 19, 14, 15, 11, 6, 15, 7, 13, 12, 9, 13, 9, 15, 17, 12, 12, 9, 7, 7, 8, 11, 14, 11, 14, 19, 15, 11, 9, 11, 11, 11, 15, 15, 17, 7, 15, 11, 16, 11, 18, 8, 10, 13, 9, 10, 13, 7, 17

Offset

2,1

Comments

a(n) is defined for n > 1.

Equal primes obtained in |P_i +- P_n / P_i| and in |P_i/2 +- 2P_n / P_i| are counted as different cases, as demonstrated in the examples.

It is conjectured that all terms of this sequence are greater than zero, but this conjecture might have exceptions.

Links

Lei Zhou, Table of n, a(n) for n = 2..500

Example

--------------------   For n = 2, a(2) = 3: --------------------
P_2 = 2*3 = 6.
P_2 + P_2 / P_2 = 6 + 1 = 7 is prime (case 1);
|P_2 - P_2 / P_2| = |6 - 1| = 5 is prime (case 2);
P_2 / 2 + 2P_2 / P_2 = 6/2 + 2 = 5 is prime (case 3);
|P_2 / 2 - 2P_2 / P_2| = |6/2 - 2| = 1 is not prime;
three ways found, so a(2) = 3.
--------------------  For n = 3, a(3) = 7: --------------------
P_2 = 2 * 3 = 6; P_3 = 2 * 3 * 5 = 30.
P_2 + P_3 / P_2 = 6 + 5 = 11 is prime (case 1);
|P_2 - P_3 / P_2| = |6 - 5| = 1 is not prime;
P_2 / 2 + 2P_3 / P_2 = 6/2 + 2 * 30/6 = 3 + 10 = 13 is prime (case 2);
|P_2 / 2 - 2P_3 / P_2| = |6/2 - 2 * 30/6| = |3 - 10| = 7 is prime (case 3);
P_3 + P_3 / P_3 = 30 + 1=31 is prime (case 4);
|P_3 - P_3 / P_3| = |30 - 1| = 29 is prime (case 5);
P_3 / 2 + 2P_3 / P_3 = 30/2 + 2 * 30/30 = 15 + 2 = 17 is prime (case 6);
|P_3 / 2 - 2P_3 / P_3| = |30/2 - 2 * 30/30| = |15 - 2| = 13 is prime (case 7);
seven ways found, so a(3) = 7.

Mathematica

Table[ct = 0; p1 = 1; Do[p1 = p1*Prime[i], {i, 2, n}]; p2 = p1; Do[p2 = p2/Prime[i]; p3 = p1/p2; If[PrimeQ[2*p2 + p3], ct++]; If[PrimeQ[Abs[2*p2 - p3]], ct++]; If[PrimeQ[p2 + 2*p3], ct++]; If[PrimeQ[Abs[p2 - 2*p3]], ct++], {i, 2, n}]; ct, {n, 2, 78}]

Crossrefs

Cf. A000040, A002110.

Sequence in context: A324774 A213687 A226934 * A308169 A343225 A363245

Adjacent sequences: A238503 A238504 A238505 * A238507 A238508 A238509

Keyword

nonn,hard

Author

Lei Zhou, Feb 27 2014

Status

approved