A204539 -id:A204539 - cp's OEIS frontend

A204540 Numbers k whose basins (as defined in A204539) equal 1.

2, 3, 4, 6, 14, 1760, 39046

Offset: 1

Colm Fagan, Jan 16 2012

Only seven such numbers are known. Referring to A204539, we cannot be certain that these are the only numbers k with the property basin(k) = 1, but there are no others for k less than 12,000,000. Furthermore the average value of basin(k) is approximately equal to k/3 for large values of k, which makes it extremely unlikely that another one will ever be found.

For unknown reasons, all integers > 5 with basin equal to 1 or 2 are even. See A185001 for the list of known numbers with basin = 2.

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.

Cf. A204539, A185001.

(* First run the program for A204539 to define basin *) Select[Range[1000], basin[#] == 1 &] (* Alonso del Arte, Jan 19 2012 *)

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

Colm Fagan, Jan 23 2012

This sequence follows on from A204540, which lists seven values of k for which basin(k) = 1. There are 37 known values of k for which basin(k) = 2. A search of numbers up to 10,000,000 has not uncovered any further integers with this property. The possibility that there are even larger numbers with basins equal to 2 cannot be completely ruled out, but the chances of one being discovered are remote, in view of the fact that the average basin size for large values of k is approximately k/3, that is, over 2000000 in the region where the last known such integer was discovered.

For unknown reasons, all integers > 5 with basins equal to 1 or 2 are even.

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.

Cf. A204539, A204540.

A104738 Positions of records in A104706.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 15, 17, 21, 24, 29, 30, 39, 41, 51, 54, 59, 66, 75, 77, 87, 96, 105, 107, 120, 129, 137, 141, 161, 165, 180, 186, 201, 209, 221, 227, 249, 255, 270, 285, 306, 311, 324, 336, 359, 366, 390, 401, 420, 435, 459, 465, 495, 501, 527, 534

Offset: 1

Zak Seidov, Mar 21 2005

nonn

The entries in this sequence are the same as the partial sums of the independently derived A204539, for reasons unknown at present. - Colm Fagan, Jan 23 2012

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A104706, A104737, A104739, A204539.

See A002491 for a conjectured connection to this sequence.

n=n+1; temp1=n^2; for k=(n-1) step -1 to 2; temp2=int(temp1/k); temp1=k*temp2; if int((temp2+k)/2))*2<>(temp2+k) then temp1=temp1-k; next k; a(n-1)=temp1/4 ' Colm Fagan, Nov 08 2015

function [ A ] = A104738( N )
% Produce a(1) : a(N)
M = N;
R = [1:M];
v = 1;
A = 1;
count = 1;
n = 1;
while count < N
    n = n+1;
    if 2*R(1)+1 > M
        R = [R, (M+1):M+N];
    end
    R = [R(2:2*R(1)+1), R(1), R((2*R(1)+2) : M)];
    if R(1) > v
        A = [A, n];
        v = R(1);
        count = count+1;
    end
end
end;
A104738(100)

A104706 = NestList[Rest[Insert[#, #[[1]], 2 + 2 #[[1]]]]&, Range[m = 1000], m][[All, 1]];
rec = 0; Reap[For[k = 1, k <= Length[A104706], k++, ak = A104706[[k]]; If[ak > rec, rec = ak; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Apr 11 2019, after Birkas Gyorgy in A104706 *)

a(n) = {n++; temp1 = n^2; forstep (k= n-1, 2, -1, temp2 = temp1\k; temp1 = k*temp2; if (((temp2+k)\2)*2 != (temp2+k), temp1 -= k)); temp1/4;} \\ after Basic; Michel Marcus, Dec 04 2015

Extended by Ray Chandler, Jan 19 2012

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A204540 Numbers k whose basins (as defined in A204539) equal 1.

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A185001 Numbers k with the property that their basins (as defined in A204539) are 2.

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A104738 Positions of records in A104706.

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