A276194 -id:A276194 - cp's OEIS frontend

A276982 a(n) = number of primes p whose balanced ternary representation is compatible with the binary representation of A276194(n).

4, 6, 8, 10, 10, 10, 7, 19, 18, 16, 19, 17, 16, 11, 20, 19, 21, 22, 21, 19, 30, 21, 22, 23, 30, 22, 30, 30, 30, 7, 24, 27, 23, 28, 24, 29, 45, 25, 29, 20, 53, 28, 50, 45, 50, 30, 24, 25, 48, 25, 45, 40, 45, 26, 53, 48, 53, 45, 50, 45, 10, 27, 26, 32, 24, 26

Offset: 1

Lei Zhou, Oct 20 2016

Let B = binary representation of A276194(n), and let C = C(p) = balanced ternary (bt) representation of a prime p (see A117966). Thus C is a string of 0's, 1's, and -1's. We will write T instead of -1.
We say that C is compatible with B if (i) length(C) = length(B); (ii) C has a 1 or T wherever B has a 1; and (iii) there is exactly one 1 or T in C in the positions where B is 0, and otherwise C has a 0 whenever B has a 0.
Then a(n) is the number of primes p for which C(p) is compatible with B.
It is conjectured that all a(n) > 0.  This has been checked for n <= 100000. But it is possible that there is a counterexample for very large n.

			n=1, A276194(1) = 5, or 101 in binary form. Using this as mask to generate positive balanced ternary numbers that allow 1 or T on all 1 digits, but only one digits 1 or T falls on a 0 digits, the following balanced ternary numbers can be generated: 1TT=5, 1T1=7, 11T=11, 111=13. All the four numbers are primes.  So a(1)=4.
n=2, A276194(2) = 9, or 1001 in binary form. Using this as mask to generate positive balanced ternary numbers that allow 1 or T on all 1 digits, but only one digits 1 or T falls on a 0 digits, the following balanced ternary numbers can be generated: 1T0T=17, 1T01=19, 10TT=23, 10T1=25, 101T=29, 1011=31, 110T=35, 1101=37.  Among the 8 numbers, 6 of them (17, 19, 23, 29, 31, and 37) are primes.  So a(2)=6.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A276194.

BNDigits[m_Integer] := Module[{n = m, d, t = {}},
   While[n > 0, d = Mod[n, 2]; PrependTo[t, d]; n = (n - d)/2]; t];
c = 1;
Table[ While[c = c + 2; d = BNDigits[c]; ld = Length[d];
   c1 = Total[d]; !(EvenQ[c1] && (c1 < ld))];
  l = Length[d]; flps = Flatten[Position[Reverse[d], 1]] - 1;
  flps = Delete[flps, Length[flps]];
  sfts = Flatten[Position[Reverse[d], 0]] - 1; lf = Length[flps]; ls = Length[sfts]; ct = 0;
  Do[Do[cp10 = 3^(l - 1) + 3^(sfts[[i]]);
    cp20 = 3^(l - 1) - 3^(sfts[[i]]); di = BNDigits[j];
    While[Length[di] < lf, PrependTo[di, 0]]; Do[
     If[di[[k]] == 0, cp10 = cp10 - 3^flps[[k]];
      cp20 = cp20 - 3^flps[[k]], cp10 = cp10 + 3^flps[[k]];
      cp20 = cp20 + 3^flps[[k]]], {k, 1, lf}];
    If[PrimeQ[cp10], ct++]; If[PrimeQ[cp20], ct++], {j, 0, 2^lf - 1}], {i, 1, ls}]; ct, {n, 1, 66}]

Edited by N. J. A. Sloane, Nov 05 2016

cp's OEIS Frontend

A276982 a(n) = number of primes p whose balanced ternary representation is compatible with the binary representation of A276194(n).

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