OFFSET
2,4
COMMENTS
The "basin" (analogous to river basins, for reasons set out below) is the number of positive integers N=4k which end in the "sea" at n^2. The "sea" of N is found as follows:
Starting out with N, in step i=1,2,3,..., stop if you have reached N=(i+1)^2 (the "sea" of N), otherwise set N to the next higher, odd or even (according to the parity of i), multiple of i+2, and go to step i+1.
Partial sums of this sequence appear to be A104738 (with a shift in offset). This has been confirmed for at least the first 4000 terms, but it is not at all clear why this is the case. - Ray Chandler, Jan 20 2012
LINKS
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.
EXAMPLE
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}
etc.
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.
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Colm Fagan, Jan 16 2012
STATUS
approved