OFFSET
1,1
COMMENTS
The only odd term is 3; the only other term not ending in 10, 30, 70, or 90 in decimal is 42.
All terms t in row m have A001221(t) = m and at least one prime q coprime to t such that q < A006530(t).
Consider "tier" m and primorial p_m# = A002110(m), let "distension" i = pi(A006530(T(m, k))) - m and let "depth" j = m - pi(A053669(T(m, k))) + 1. Distension is the difference in the index of gpf(T(m, k)) and pi(m), while depth is the difference between the index of the least prime totative of T(m, k) and pi(m) + 1. We can calculate the maximum distension i given m and j via i_max = A020900(m - j + 1) - m - j + 1. This enables us to use permutations of 0 and 1 values in the notation A054841 and produce a(n) with some efficiency.
The most efficient method of generating a(n) is via f(x) = A287352(x), i.e., subtracting 1 from all values in row x of A287352. We use a pointer variable to direct increment on f(p_m#) = a constant array of m 1's, until we have exhausted producing terms p_m# < t < 2*p_m#. This enables the generation of T(m, k) for 1 <= m <= 100.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..14936 (Rows 1 <= m <= 36)
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Squarefree
Michael De Vlieger, Relations between A288813, A288784, A002110, and A244052, including prime decompositions of terms of a(n) and all code used to generate the tables.
EXAMPLE
Triangle begins:
n a(n)
1: 3
2: 10
3: 42
4: 330 390
5: 2730 3570 3990 4290
6: 39270 43890 46410 51870 53130
7: 570570 690690 746130 870870 881790 903210 930930 1009470
...
MATHEMATICA
Table[Function[P, Select[Range[P + 1, 2 P - 1], And[SquareFreeQ@ #, PrimeOmega@ # == n] &]]@ Product[Prime@ i, {i, n}], {n, 7}] // Flatten (* Michael De Vlieger, Jun 24 2017 *)
f[n_] := Block[{P = Product[Prime@ i, {i, n}], lim, k = 1, c, w = ConstantArray[1, n]}, lim = 2 P; Sort@ Reap[Do[w = If[k == 1, MapAt[# + 1 &, w, -k], Join[Drop[MapAt[# + 1 &, w, -k], -k + 1], ConstantArray[1, k - 1]]]; c = Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ w]; If[c < lim, Sow[c]; k = 1, If[k == n, Break[], k++]], {i, Infinity}] ][[-1, 1]] ]; Array[f, 9] // Flatten (* Michael De Vlieger, Jun 28 2017, faster *)
PROG
(PARI) primo(n) = prod(i=1, n, prime(i));
row(n) = my(vrow = []); for (j=primo(n)+1, 2*primo(n)-1, if (issquarefree(j) && (omega(j)==n), vrow = concat(vrow, j))); vrow;
tabf(nn) = for (n=1, nn, print(row(n))); \\ Michel Marcus, Jun 29 2017
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Michael De Vlieger, Jun 24 2017
STATUS
approved