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%I #18 Jul 15 2017 11:05:41
%S 3,10,42,330,390,2730,3570,3990,4290,39270,43890,46410,51870,53130,
%T 570570,690690,746130,870870,881790,903210,930930,1009470,11741730,
%U 13123110,14804790,15825810,16546530,17160990,17687670,18888870,281291010,300690390,340510170
%N Irregular triangle read by rows: T(m, k) is the list of squarefree numbers A002110(m) < t < 2*A002110(m) such that A001221(t) = m.
%C a(n) = terms t of row m of A288784 such that A002110(m) < t < 2*A002110(m).
%C The only odd term is 3; the only other term not ending in 10, 30, 70, or 90 in decimal is 42.
%C All terms t in row m have A001221(t) = m and at least one prime q coprime to t such that q < A006530(t).
%C 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.
%C 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.
%H Michael De Vlieger, <a href="/A288813/b288813.txt">Table of n, a(n) for n = 1..14936</a> (Rows 1 <= m <= 36)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Primorial.html">Primorial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H Michael De Vlieger, <a href="/A288813/a288813.txt">Relations between A288813, A288784, A002110, and A244052</a>, including prime decompositions of terms of a(n) and all code used to generate the tables.
%e Triangle begins:
%e n a(n)
%e 1: 3
%e 2: 10
%e 3: 42
%e 4: 330 390
%e 5: 2730 3570 3990 4290
%e 6: 39270 43890 46410 51870 53130
%e 7: 570570 690690 746130 870870 881790 903210 930930 1009470
%e ...
%t 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 *)
%t 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 *)
%o (PARI) primo(n) = prod(i=1, n, prime(i));
%o row(n) = my(vrow = []); for (j=primo(n)+1, 2*primo(n)-1, if (issquarefree(j) && (omega(j)==n), vrow = concat(vrow, j))); vrow;
%o tabf(nn) = for (n=1, nn, print(row(n))); \\ _Michel Marcus_, Jun 29 2017
%Y Cf. A001221, A002110, A020900, A288784.
%K nonn,tabf,easy
%O 1,1
%A _Michael De Vlieger_, Jun 24 2017
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