A288784 - OEIS

%I #24 Jun 24 2017 06:50:15

%S 1,2,3,4,6,10,12,18,24,30,42,60,84,90,120,150,180,210,330,390,420,630,

%T 840,1050,1260,1470,1680,1890,2100,2310,2730,3570,3990,4290,4620,5460,

%U 6930,8190,9240,10920,11550,13650,13860,16170,18480,20790,23100,25410,27720

%N Irregular triangle read by rows: T(n,m) is the list of numbers k*A002110(n) <= k*t < (k + 1)*A002110(n) such that A001222(k*t) = n, with 1 <= k < prime(n + 1).

%C A060735 and A002110 are subsets.

%C This sequence is a necessary but insufficient condition for A244052. Terms that are in A060735 and A002110 are also in A244052. The first terms of this sequence that are not in A244052 are {3, 4290, 881790, 903210, 1009470, 17160990, 363993630, 380570190, 406816410, 434444010, ...}.

%C Primorial p_n# = A002110(n) is the smallest squarefree number with n prime factors. Consider the list of squarefree numbers t with n prime factors greater than and including A002110(n) but less than 2*A002110(n). Extend the list to include products k*t of this list with 1 <= k < prime(n+1) such that k*t < (k+1)*p_n#. This list contains squarefree numbers k*t with n distinct primes and presumes that the number (k+1)*p_n# serves as a "limit" beyond which k*t > (k+1)p_n# are not in the sequence.

%H Michael De Vlieger, <a href="/A288784/b288784.txt">Table of n, a(n) for n = 0..11793</a> (rows 1 <= n <= 30).

%H Michael De Vlieger, <a href="/A288784/a288784.txt">Terms of a(n) with 1 <= n <= 653 in A002110, A060735, and A244052</a>

%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>

%e Triangle begins:

%e     n   T(n,m)

%e     0:   1;

%e     1:   2,  3,  4;

%e     2:   6, 10, 12, 18, 24;

%e     3:  30, 42, 60, 84, 90, 120, 150, 180;

%e        ...

%t Table[Function[P, Function[s, Flatten@ Map[Function[k, Select[k s, # < (k + 1) P &]], Range[1, Prime[n + 1] - 1]]]@ Select[Range[P, 2 P - 1], And[SquareFreeQ@ #, PrimeOmega@ # == n] &]]@ Product[Prime@ i, {i, n}], {n, 0, 5}] (* _Michael De Vlieger_, Jun 15 2017 *)

%Y Cf. A001221, A002110, A005117, A006881, A033992, A060735, A244052.

%K nonn,tabf

%O 0,2

%A _Michael De Vlieger_, Jun 15 2017