A182855 Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map.

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Author

Matthew Vandermast, Jan 04 2011

Keywords

• nonn

Comments

In each case, 2 is the fixed point that is reached (1 is the other fixed point of the x -> A181819(x) map).

Includes all integers whose prime signature a) contains two or more distinct numbers, and b) contains no number that occurs the same number of times as any other number. The first member of this sequence that does not fit that description is 75675600, whose prime signature is (4,3,2,2,1,1).

A full characterization is: Numbers whose prime signature (1) has not all equal multiplicities but (2) the numbers of distinct parts appearing with each distinct multiplicity are all equal. For example, the prime signature of 2520 is {1,1,2,3}, which satisfies (1) but fails (2), as the numbers of distinct parts appearing with each distinct multiplicity are 1 (with multiplicity 2, the part being 1) and 2 (with multiplicity 1, the parts being 2 and 3). Hence the sequence does not contain 2520. - Gus Wiseman, Jan 02 2019

Examples

			1. 180 requires exactly five iterations under the x -> A181819(x) map to reach a fixed point (namely, 2).  A181819(180) = 18;  A181819(18) = 6; A181819(6) = 4; A181819(4) = 3;  A181819(3) = 2 (and A181819(2) = 2).
2. The prime signature of 180 (2^2*3^2*5) is (2,2,1).
a. Two distinct numbers appear in (2,2,1) (namely, 1 and 2).
b. Neither 1 nor 2 appears in (2,2,1) the same number of times as any other number that appears there.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Fixed Point
• Eric Weisstein's World of Mathematics, Map

Crossrefs

Numbers n such that A182850(n) = 5. See also A182853, A182854.

Subsequence of A059404 and A182851. Includes A085987 and A179642 as subsequences.

Cf. A001221, A001222, A006939, A062770, A071625, A118914, A181819, A181821, A182857, A323014, A323022, A323023.

Programs

Mathematica

Select[Range[1000],With[{sig=Sort[Last/@FactorInteger[#]]},And[!SameQ@@Length/@Split[sig],SameQ@@Length/@Union/@GatherBy[sig,Length[Position[sig,#]]&]]]&] (* Gus Wiseman, Jan 02 2019 *)