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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).
1
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.
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
Sequence in context: A324774 A213687 A226934 * A308169 A343225 A363245
KEYWORD
nonn,hard
AUTHOR
Lei Zhou, Feb 27 2014
STATUS
approved