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Irregular triangle read by rows: T(n,k) is the number of primes with n balanced ternary digits of which 2k+1 (3 <= 2k+1 <= n) are nonzero.
0

%I #23 Nov 10 2019 14:40:44

%S 4,6,8,10,7,35,11,70,30,7,129,143,10,191,458,93,11,262,1112,605,11,

%T 370,2209,2513,273,8,484,4007,7646,2562,10,595,6683,19361,12878,938,9,

%U 765,10697,42633,47555,10311,11,917,16103,85860,143382,62541,3183

%N Irregular triangle read by rows: T(n,k) is the number of primes with n balanced ternary digits of which 2k+1 (3 <= 2k+1 <= n) are nonzero.

%C This sequence has the same indexing rule as A277513.

%C There are no zeros in this sequence up to the (10^5)-th term.

%C It is conjectured that all terms of this sequence are greater than zero, or else there are infinitely many zero terms. The first zero term might appear beyond 10^6.

%e When n=3 and k=1, there are the following three trits balanced ternary numbers: 5=1TT, 7=1T1, 11=11T, 13=111. All four of them are primes, so T(3,1) = 4;

%e When n=4 and k=1, there are the following balanced ternary numbers with 2k+1=3 nonzero trits: 17=1T0T, 19=1T01, 23=10TT, 25=10T1, 29=101T, 31=1011, 35=110T, 37=1101. Among these 8 numbers, 6 of them are prime, so T(4,1) = 6.

%e By listing the first few rows, this sequence appears as:

%e k=1 2 3 4

%e n=3 4

%e n=4 6

%e n=5 8 10

%e n=6 7 35

%e n=8 11 70 30

%e n=9 7 129 143

%e n=10 10 191 458 93

%t (* This converts number m to balanced ternary form, stores the result in list t. *)

%t BTDigits[m_Integer, g_] :=

%t Module[{n = m, d, sign, t = g},

%t If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n];

%t d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++];

%t While[Length[t] < d, PrependTo[t, 0]];

%t t[[Length[t] + 1 - d]] = sign;

%t t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];

%t (* This calculates j and k for balanced ternary form of number m. *)

%t BTnonzeroNumofDigits[m_Integer] :=

%t Module[{n = m}, t = BTDigits[n, {}]; j = Length[t];

%t k = 0; Do[If[t[[i]] != 0, k++], {i, 1, j}];

%t k = (k - 1)/2; {j, k}];

%t (* This calculates the category index n as defined in A277513 for a {j,k} pair. *)

%t IndexA277513[{j_, k_}] :=

%t Module[{m, i},

%t If[OddQ[j], m = (j - 1)/2; i = m^2 - m + k, m = j/2;

%t i = m^2 - 2 m + 1 + k]];

%t (* This counts a(n). *)

%t p=3;a={} ;While[p = NextPrime[p]; jk = BTnonzeroNumofDigits[p]; jk[[1]] <= 15, id = IndexA277513[jk]; While[Length[a] < id, AppendTo[a, 0]];

%t a[[id]]++];a

%Y 2 together with the column 1 gives A196698.

%Y Cf. A277513, A134023.

%K nonn,base,tabf

%O 3,1

%A _Lei Zhou_, Oct 18 2016