A380473 Numbers k neither squarefree nor prime power (i.e., in A126706) such that A119288(k) <= A003557(k) < A053669(k) < A006530(k).

126, 168, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 666, 696, 738, 744, 774, 846, 888, 954, 984, 990, 1032, 1062, 1098, 1128, 1170, 1206, 1272, 1278, 1314, 1320, 1386, 1416, 1422, 1464, 1494, 1530, 1560, 1602, 1608, 1638, 1650, 1704, 1710, 1746

Offset: 1

Author

Michael De Vlieger, Jul 22 2025

Keywords

• nonn

Comments

Let rad = A007947, p = A119288, q = A053669, g = A006530, and r = A003557.

Numbers k in A126706 such that p <= r < q < g.

Terms are products k of a number s in A033845 and a number t in A007310 with at least one prime power factor p^m | k such that m > 1.

Examples

			Table of n, a(n) for select n:
   n    a(n)                       r   q
  --------------------------------------
   1    126 = 2 * 3^2 * 7          3   5
   2    168 = 2^3 * 3 * 7          4   5
   3    198 = 2 * 3^2 * 11         3   5
   4    234 = 2 * 3^2 * 13         3   5
   5    264 = 2^3 * 3 * 11         4   5
   6    306 = 2 * 3^2 * 17         3   5
   7    312 = 2^3 * 3 * 13         4   5
  24    990 = 2 * 3^2 * 5 * 11     3   7
  29   1170 = 2 * 3^2 * 5 * 13     3   7
  45   1650 = 2 * 3 * 5^2 * 11     5   7
  57   1980 = 2^2 * 3^2 * 5 * 11   6   7
  68   2340 = 2^2 * 3^2 * 5 * 13   6   7
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003557, A007310, A007947, A033845, A053669, A055932, A080259, A119288, A126706, A364998, A369540.

Programs

Mathematica

a053669[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q] ]; q];
s = Select[Range[2^12], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
Select[s, And[#3 < #4 < #2[[-1, 1]], #2[[2, 1]] <= #3] & @@
  {#1, #2, #1/Apply[Times, #2[[All, 1]]], a053669[#1]} & @@
  {#, FactorInteger[#]} &]

Formula

Intersection of A364998 and A080259 = A364998 \ A055932 = A364998 \ A369540.