A193586 Number of attractors under iteration of sum of squares of digits in base n.
1, 5, 1, 6, 9, 13, 10, 8, 9, 9, 20, 13, 12, 35, 7, 15, 7, 21, 27, 37, 24, 36, 32, 26, 10, 36, 27, 28, 10, 56, 22, 26, 23, 63, 39, 27, 19, 67, 9, 36, 40, 54, 54, 48, 18, 73, 52, 75, 18, 117, 52, 74, 22, 65, 48, 53, 45, 44, 43, 18, 30, 67, 39, 49, 87, 111, 15
Offset: 2
A377087 Number of cycles 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.
0, 1, 0, 1, 2, 1, 2, 0, 3, 2, 5, 2, 4, 1, 3, 3, 3, 2, 3, 4, 1, 5, 3, 7, 2, 3, 3, 3, 4, 4, 5, 3, 6, 12, 2, 1, 3, 2, 6, 10, 4, 8, 6, 3, 4, 2, 3, 1, 3, 4, 9, 3, 2, 2, 5, 7, 4, 8, 7, 5, 6, 6, 6, 1, 8, 7, 4, 6, 6, 2, 5, 7, 5, 5, 4, 5, 3, 4, 3, 5, 2, 4, 7, 8, 3, 7, 7
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 and three cycles. Therefore a(10) = 3.
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.
A336744 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly three.
14, 66, 94, 300, 384, 436, 496, 750, 1406, 1794, 2336, 2624, 28034
Offset: 1
Comments
Let b > 1 be an integer, and write the base b expansion of any nonnegative integer m as m = x_0 + x_1 b + ... + x_d b^d with x_d > 0 and 0 <= x_i < b for i=0,...,d.
Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(m) := x_0^2+ ... + x_d^2.
This is the 'sum of the squares of the digits' dynamical system alluded to in the name of the sequence.
It is known that the orbit set {m,S_{x^2,b}(m), S_{x^2,b}(S_{x^2,b}(m)), ...} is finite for all m>0. Each orbit contains a finite cycle, and for a given base b, the union of such cycles over all orbit sets is finite. Let us denote by L(x^2,i) the set of bases b such that the set of cycles associated to S_{x^2,b} consists of exactly i elements. In this notation, the sequence is the set of known elements of L(x^2,3).
A 1978 conjecture of Hasse and Prichett describes the set L(x^2,2). New elements have been added to this set in the paper Integer Dynamics, by D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang. It is natural to wonder whether the set L(x^2,3) is infinite. It is a folklore conjecture that L(x^2,1) = {2,4}.
Examples
For instance, in base 14, the three cycles are (1), (37,85), and (25,122,164,221,123,185,178,244,46). To verify that (37,85) is a cycle in base 14, note that 37=9+2*14, and that 9^2+2^2=85. Similarly, 85=1+6*14, and 1^2+6^2=37.
Links
- H. Hasse and G. Prichett, A conjecture on digital cycles, J. reine angew. Math. 298 (1978), 8--15. Also on GDZ.
- D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang, Integer Dynamics, preprint.
Crossrefs
Formula
A336762 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly two.
6, 10, 16, 20, 26, 40, 8626, 481360
Offset: 1
Comments
Let b > 1 be an integer, and write the base b expansion of any nonnegative integer n as n = x_0 + x_1 b + ... + x_d b^d with x_d > 0 and 0 <= x_i < b for i=0,...,d.
Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(n) := x_0^2+ ... + x_d^2.
It is known that the orbit set {n, S_{x^2,b}(n), S_{x^2,b}(S_{x^2,b}(n)), ...} is finite for all n > 0. Each orbit contains a finite cycle, and for a given b, the union of such cycles over all orbit sets is finite. Let us denote by L(x^2,i) the set of bases b such that the set of cycles associated to S_{x^2,b} consists of exactly i elements. In this notation, the sequence is the set of known elements of L(x^2,2).
A 1978 conjecture of Hasse and Prichett describes the set L(x^2,2). New elements have been added to this set in the paper Integer Dynamics, by D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang. The sequence contains all b <= 10^6 that are in L(x^2,2).
Examples
For instance, b = 10 is in this sequence since in the decimal system, there are exactly two cycles (1) and (4, 16, 37, 58, 89, 145, 42, 20).
Links
- H. Hasse and G. Prichett, A conjecture on digital cycles, J. reine angew. Math. 298 (1978), 8--15. Also on GDZ.
- D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang, Integer Dynamics, preprint.
Comments
Examples
Links
Crossrefs
Programs
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 b^2 do Q:=S(i,2,b); A:=[op(A),Q[nops(Q)]]; od: return(nops({op(A)})); end: seq(NumberOfAttractors(b),b=2..50);