A193585 Number of cycles under iteration of sum of squares of digits in base b.
0, 1, 0, 1, 1, 2, 3, 2, 1, 2, 4, 3, 2, 7, 1, 2, 1, 3, 1, 6, 2, 8, 4, 6, 1, 5, 4, 6, 2, 8, 6, 5, 3, 5, 4, 5, 3, 6, 1, 7, 6, 6, 2, 5, 4, 11, 4, 4, 4, 6, 3, 11, 4, 9, 4, 8, 4, 6, 6, 5, 4, 9, 6, 5, 2, 6, 3, 7, 7, 8, 5, 14, 5, 8, 3, 6, 3, 4, 5, 10, 5, 10, 6, 8, 5
Offset: 2
A377088 Number of attractors under iteration of the map sending a positive integer to the product of its leading base-n digit and the sum of the squares of its base-n digits.
1, 5, 2, 3, 8, 6, 11, 4, 16, 14, 23, 18, 42, 7, 24, 34, 26, 58, 98, 51, 99, 88, 51, 57, 103, 72, 89, 60, 69, 35, 78, 146, 39, 90, 73, 11, 109, 113, 71, 156, 220, 93, 176, 101, 132, 172, 187, 10, 160, 95, 221, 226, 69, 55, 163, 110, 137, 287, 168, 69, 260, 194, 208
Offset: 2
Comments
If b>=2 and a>=b^3 then E(a,2,b)
Examples
In the decimal system all integers go to (1), (298), (46, 208, 136), (26, 80, 512, 150), or (33, 54, 205, 58, 445, 228, 144) under iteration of the map A376270, hence there are two fixed points, one 3-cycle, one 4-cycle, and one 7-cycle. Therefore a(10) = 1 + 1 + 3 + 4 + 7 = 16.
Links
- Nathan Fox, Table of n, a(n) for n = 2..100
- N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.
A193594 Number of attractors under iteration of sum of cubes of digits in base b.
1, 6, 9, 6, 9, 34, 11, 28, 15, 46, 22, 50, 49, 60, 86, 86, 60, 128, 22, 58, 118, 93, 64, 185, 5, 109, 102, 100, 122, 184, 51, 94, 205, 131, 173, 275, 67, 216, 131, 127, 34, 360, 114, 78, 215, 213, 393, 479, 76, 254, 634, 197, 214, 496, 348, 170, 437, 349, 290
Offset: 2
Comments
If b>=2 and a >= 2*b^3, then S(a,3,b)
Examples
In the decimal system all integers go to (1); (153); (370); (371); (407) or (55, 250,133); (136, 244); (160, 217, 352); (919, 1459) under the iteration of sum of cubes of digits, hence there are five fixed points, two 2-cycles and two 3-cycles. Therefore a(10) = 5 + 2*2 + 2*3 = 15.
Links
- H. G. Grundman, E. A. Teeple, Generalized Happy Numbers, Fibonacci Quarterly 39 (2001), nr. 5, p. 462-466.
Crossrefs
Cf. A193586.
Programs
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Maple
S:=proc(n, p, b) local Q, k, N, z; Q:=[convert(n, base, b)]; for k from 1 do N:=Q[k]; z:=convert(sum(N['i']^p, 'i'=1..nops(N)), base, b); if not member(z, Q) then Q:=[op(Q), z]; else Q:=[op(Q), z]; break; fi; od; return Q; end: NumberOfAttractors:=proc(b) local A,i,Q; A:=[]: for i from 1 to 2*b^3 do Q:=S(i,3,b); A:=[op(A),Q[nops(Q)]]; od: return(nops({op(A)})); end: seq(NumberOfAttractors(b),b=2..20);
Comments
Examples
Links
Crossrefs