Colm Fagan has authored 3 sequences.
A185001
Numbers k with the property that their basins (as defined in A204539) are 2.
Original entry on oeis.org
5, 8, 10, 16, 22, 26, 106, 110, 182, 234, 282, 288, 318, 434, 766, 1056, 1072, 1462, 1550, 1930, 3024, 4330, 5424, 9398, 10634, 53094, 90602, 151632, 384002, 511638, 530102, 1364850, 1887006, 2193072, 3138096, 6470672, 6959070
Offset: 1
- Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, Journal of Integer Sequences, Vol. 24 (2021), Article 21.7.1.
- Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, arXiv:2202.02381 [math.NT], 2022.
A204540
Numbers k whose basins (as defined in A204539) equal 1.
Original entry on oeis.org
2, 3, 4, 6, 14, 1760, 39046
Offset: 1
- Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, Journal of Integer Sequences, Vol. 24 (2021), Article 21.7.1.
- Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, arXiv:2202.02381 [math.NT], 2022.
A204539
a(n) is the number of integers N=4k whose "basin" sequence (cf. comment) ends in n^2.
Original entry on oeis.org
1, 1, 1, 2, 1, 3, 2, 4, 2, 4, 3, 5, 1, 9, 2, 10, 3, 5, 7, 9, 2, 10, 9, 9, 2, 13, 9, 8, 4, 20, 4, 15, 6, 15, 8, 12, 6, 22, 6, 15, 15, 21, 5, 13, 12, 23, 7, 24, 11, 19, 15, 24, 6, 30, 6, 26, 7, 27, 26, 13, 6, 33, 27, 30, 5, 13, 30, 30, 5, 37, 15, 26, 28, 32, 7, 17, 25, 54, 9, 30, 21, 41, 25
Offset: 2
For integers N=4,8,12,16,... we have the following sequences:
{4}
{8, 9} (8 -> the next higher odd multiple of 3, which is 9 -> STOP)
{12, 15, 16} (12 -> 3*5=15 -> 4*4=16 -> STOP)
{16, 21, 24, 25}
{20, 21, 24, 25}
{24, 27, 32, 35, 36}
{28, 33, 40, 45, 48, 49}
{32, 33, 40, 45, 48, 49}
{36, 39, 40, 45, 48, 49}
...
Thus there is 1 integer N=4k ending in the sea at 2^2, whence basin a(2)=1, and idem for 3 and 4.
The two integers 16 and 20 end at 5^2, so the basin of 5 is a(5)=2.
There is again a(6)=1 integer ending in 6^2, while the basin of 7 are the 3 integers 28, 32, and 36, which all merge into the "river" that enters the "sea" in 7^2=49.
Thus the first 6 terms in the sequence are 1, 1, 1, 2, 1, 3.
Take N=100 as an example: the next integer on the same line is the next higher odd multiple of 3, i.e., smallest 3*(2m+1) > 100, which is 105. The next number is the least even multiple of 4, 4*(2m) = 112, etc., leading to 115 = 5*(2m+1), followed by 120 = 6*(2m), 133 = 7*(2m+1), 144 = 8*2m (where we have a square, but not the square of 8), 153 =9*(2m+1), 160 = 10*2m, 165 = 11*(2m+1), 168 = 12*(2m) and finally 169 = 13*13.
- Ray Chandler, Table of n, a(n) for n = 2..10001
- Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, Journal of Integer Sequences, Vol. 24 (2021), Article 21.7.1.
- Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, arXiv:2202.02381 [math.NT], 2022.
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cumul[n_Integer] := Module[{den1 = n, num = n^2, den2}, While[num > 4 && den1 != 2, num = num - 1; den1 = den1 - 1; den2 = Floor[num/den1]; If[Not[EvenQ[den1 + den2]], den2 = den2 - 1]; num = den1 den2]; Return[num/4]]; basin[2] := 1; basin[n_Integer] := cumul[n] - cumul[n - 1]; Table[basin[n], {n, 2, 75}] (* Alonso del Arte, Jan 19 2012 *)
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bs(n,s,m=2)={while(n>m^2,n=(n\m+++2-bittest(n\m-m,0))*m; s & print1(n","));n}
n=4; for(c=2,50, for(k=1,9e9, bs(n+=4)==c^2 || print1(k",")||break)) \\ M. F. Hasler, Jan 20 2012
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