[seqfan] Re: As much as I hate "base" sequences...

[Hans Havermann]

Neil: "Hans, you said: I found that the size of S fluctuates wildly, never exceeding 120 and dropping to 0 for A235601(441). Me: what is this sequence?!!"

{8, 14, 15, 17, 21, 23, 21, 22, 27, 27, 30, 32, 26, 20, 23, 25, 22, 17, 20, 22, 32, 41, 34, 35, 38, 43, 45, 49, 51, 52, 43, 44, 41, 51, 40, 38, 37, 30, 30, 35, 26, 24, 24, 30, 22, 21, 25, 21, 18, 24, 19, 25, 23, 23, 22, 31, 22, 27, 28, 25, 21, 21, 25, 28, 29, 19, 17, 15, 21, 23, 26, 28, 21, 20, 22, 23, 27, 25, 20, 20, 24, 40, 46, 50, 51, 47, 46, 47, 40, 32, 29, 31, 35, 37, 45, 41, 36, 41, 39, 35, 23, 23, 21, 20, 25, 25, 15, 12, 18, 20, 24, 33, 34, 40, 59, 63, 59, 54, 51, 48, 46, 50, 52, 51, 48, 48, 43, 46, 41, 39, 31, 35, 34, 22, 21, 11, 9, 10, 10, 17, 23, 18, 18, 19, 23, 23, 32, 27, 26, 26, 30, 34, 26, 23, 26, 25, 24, 26, 39, 43, 36, 37, 30, 31, 35, 37, 39, 34, 31, 42, 48, 51, 46, 50, 48, 53, 53, 58, 48, 40, 40, 28, 32, 42, 36, 25, 21, 22, 27, 16, 19, 13, 20, 27, 18, 11, 14, 13, 20, 21, 19, 21, 14, 13, 25, 26, 31, 39, 40, 37, 31, 37, 43, 47, 43, 45, 46, 34, 30, 18, 27, 33, 34, 40, 38, 31, 26, 30, 30, 27, 33, 44, 58, 53, 60, 67, 72, 71, 66, 60, 70, 84, 82, 93, 92, 99, 107, 107, 113, 118, 112, 120, 103, 89, 74, 63, 52, 47, 45, 44, 34, 38, 36, 41, 41, 38, 39, 46, 54, 58, 54, 61, 72, 66, 67, 60, 61, 51, 51, 44, 51, 55, 56, 66, 78, 94, 90, 89, 87, 85, 70, 57, 47, 47, 49, 53, 52, 58, 56, 64, 71, 60, 56, 56, 59, 44, 44, 34, 40, 51, 47, 44, 42, 41, 39, 38, 28, 30, 32, 36, 32, 37, 45, 49, 48, 43, 45, 39, 29, 23, 26, 22, 23, 22, 18, 14, 17, 16, 15, 15, 14, 10, 15, 13, 14, 15, 13, 19, 20, 18, 11, 9, 10, 15, 21, 20, 17, 11, 13, 13, 7, 12, 17, 18, 21, 25, 34, 44, 39, 29, 35, 36, 34, 36, 40, 32, 35, 41, 50, 49, 42, 38, 45, 39, 41, 40, 30, 30, 33, 32, 26, 21, 26, 26, 29, 23, 19, 14, 5, 8, 7, 8, 8, 10, 15, 17, 15, 16, 17, 14, 12, 8, 6, 9, 10, 8, 7, 4, 11, 10, 13, 17, 12, 9, 14, 12, 5, 13, 9, 12, 16, 17, 19, 13, 11, 8, 4, 3, 2, 4}

The first '8' is the set {2,3,4,5,6,7,8,9} which require one step each to get to one; The second '14' is the set {12,18,21,24,27,36,42,45,48,54,63,72,81,84} which require two steps and the third '15' is {108,162,216,243,324,378,405,432,486,648,756,864,972,1296,1458} which require three steps; and so on. The final (very large) 4 numbers require 440 steps, which (if I have done this correctly) appears to be the largest number of steps possible (because none of these 4 numbers apparently have a predecessor).

I don't think this count sequence is necessarily OEIS-worthy but I'll submit the finite sequence of the corresponding (total) 15094 numbers that end up as 1 under repeated applications of the A235601-defined map f.