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
KEYWORD
nonn,hard
AUTHOR
Lei Zhou, Feb 27 2014
STATUS
approved