A293728 -id:A293728 - cp's OEIS frontend

A293725 Numbers k such that c(k,0) = c(k,1), where c(k,d) = number of d's in the first k digits of the base-2 expansion of sqrt(2).

2, 10, 20, 24, 28, 32, 318, 328, 330, 334, 336, 608, 622, 636, 638, 674, 676, 678, 680, 682, 826, 828, 832, 836, 838, 842, 844, 846, 848, 850, 852, 856, 858, 876, 880, 884, 886, 898, 906, 908, 918, 920, 928, 930, 942, 944, 946, 948, 950, 962, 964, 966, 968

Offset: 1

Clark Kimberling, Oct 16 2017

This sequence together with A293727 and A293728 partition the positive integers.

			In base 2, sqrt(2) = 1.0110101000001001111001..., so that initial segments 1.0; 1.011010100..., of lengths 2,10,... have the same number of 0's and 1's.

Cf. A004539, A002103, A293726, A293727, A293728.

z = 300; u = N[Sqrt[2], z]; d = RealDigits[u, 2][[1]];
t[n_] := Take[d, n]; c[0, n_] := Count[t[n], 0]; c[1, n_] := Count[t[n], 1];
Table[{n, c[0, n], c[1, n]}, {n, 1, 100}]
u = Select[-1 + Range[z], c[0, #] == c[1, #] &]  (* A293725 *)
u/2  (* A293726 *)
Select[-1 + Range[z], c[0, #] < c[1, #] &]  (* A293727 *)
Select[-1 + Range[z], c[0, #] > c[1, #] &]  (* A293728 *)

A293726 a(n) = (1/2)*A293725(n).

Original entry on oeis.org

1, 5, 10, 12, 14, 16, 159, 164, 165, 167, 168, 304, 311, 318, 319, 337, 338, 339, 340, 341, 413, 414, 416, 418, 419, 421, 422, 423, 424, 425, 426, 428, 429, 438, 440, 442, 443, 449, 453, 454, 459, 460, 464, 465, 471, 472, 473, 474, 475, 481, 482, 483, 484

Offset: 1

Clark Kimberling, Oct 18 2017

Cf. A004539, A002103, A293726, A293727, A293728.

z = 300; u = N[Sqrt[2], z]; d = RealDigits[u, 2][[1]];
t[n_] := Take[d, n]; c[0, n_] := Count[t[n], 0]; c[1, n_] := Count[t[n], 1];
Table[{n, c[0, n], c[1, n]}, {n, 1, 100}]
u = Select[Range[z], c[0, #] == c[1, #] &]  (* A293725 *)
u/2  (* A293726 *)
Select[Range[z], c[0, #] < c[1, #] &]  (* A293727 *)
Select[Range[z], c[0, #] > c[1, #] &]  (* A293728 *)

A293727 Numbers k such that c(k,0) < c(k,1), where c(k,d) = number of d's in the first k digits of the base-2 expansion of sqrt(2).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Clark Kimberling, Oct 18 2017

This sequence together with A293725 and A293728 partition the nonnegative integers.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A004539, A002103, A293726, A293727, A293728.

z = 300; u = N[Sqrt[2], z]; d = RealDigits[u, 2][[1]];
t[n_] := Take[d, n]; c[0, n_] := Count[t[n], 0]; c[1, n_] := Count[t[n], 1];
Table[{n, c[0, n], c[1, n]}, {n, 1, 100}]
u = Select[Range[z], c[0, #] == c[1, #] &]  (* A293725 *)
u/2  (* A293726 *)
Select[Range[z], c[0, #] < c[1, #] &]  (* A293727 *)
Select[Range[z], c[0, #] > c[1, #] &]  (* A293728 *)

cp's OEIS Frontend

A293725 Numbers k such that c(k,0) = c(k,1), where c(k,d) = number of d's in the first k digits of the base-2 expansion of sqrt(2).

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A293726 a(n) = (1/2)*A293725(n).

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A293727 Numbers k such that c(k,0) < c(k,1), where c(k,d) = number of d's in the first k digits of the base-2 expansion of sqrt(2).

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