A231387 a(n) is a prime number that cannot be the center term of a length 3 arithmetic progression prime group with a common difference whose number of runs in binary expansion is 2.
2, 3, 199, 1319, 2371, 2437, 3253, 6871, 6991, 7937, 7951, 9041, 9973, 10631, 11941, 12379, 12671, 13009, 13147, 13187, 13267, 13799, 13859, 14479, 16889, 17519, 20051, 21089, 26003, 27281, 27529, 29581, 31477, 32009, 34439, 38561, 41611, 42719, 43543, 44839
Offset: 1
Examples
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.
Links
- Lei Zhou, Table of n, a(n) for n = 1..10000
Programs
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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}]
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