OFFSET
1,1
COMMENTS
Fewer than 0.7% of the first three million primes have this property.
AP-3 is defined as 3 prime groups in arithmetic progression.
Hypothesized property: for n>=3, there exists at least one number k whose number of runs in binary expansion (A005811) equals 4, such that {a(n)-k, a(n), a(n)+k} forms an AP-3 group.
LINKS
Lei Zhou, Table of n, a(n) for n = 1..10000
EXAMPLE
2 and 3 cannot be the second term of an AP-3 prime group, so a(1)=2 and a(2)=3;
For any prime numbers between 5 and 197, there exists at least one number k in A043569 such that {a-k, a, a+k} forms an AP-3 prime group. For example, when p=197, there are the following groups {5,197,389}, {101,197,293}, {137,197,257}, and {167,197,227} with corresponding k = 192 = 11000000 base 2 = A043569(23), 96 = 1100000 base 2 = A043569(17), 60 = 111100 base 2 = A043569(14), and 30 = 11110 base 2 = A043569(10).
However, when p = 199, among all six AP-3 groups, {19,199,379}, {31,199,367}, {61,199,337}, {67,199,331}, {127,199,271}, and {157,199,241}, none of k value (180 = 10110100 base 2, 168 = 10101000 base 2, 138 = 10001010 base 2, 132 = 10000100 base 2, 72 = 1001000, and 42 = 101010 respectively) is a term of A043569. None of them is in the form of 1..10..0 base 2 thus not an element of A043569.
So a(3)=199.
MATHEMATICA
seed = 1; Table[While[seed = NextPrime[seed]; sum = seed*2; lowbond = sum; cp1 = seed; While[cp1 = NextPrime[cp1]; (lowbond > 2) && (cp1 < sum), cp2 = sum - cp1; If[PrimeQ[cp2], test = cp2 - cp1; rank = Length[Length /@ Split[IntegerDigits[test, 2]]]; lowbond = Min[rank, lowbond]]]; lowbond == 2]; seed, {i, 1, 41}]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Lei Zhou, Nov 08 2013
STATUS
approved