A288813 - OEIS

%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