A114183 -id:A114183 - cp's OEIS frontend

A221715 Start of n-th doubling run in A114183.

1, 5, 3, 9, 33, 11, 13, 7, 21, 25, 20, 17, 23, 19, 49, 39, 35, 47, 27, 29, 15, 43, 37, 97, 55, 41, 51, 57, 85, 73, 193, 111, 59, 61, 31, 89, 53, 329, 145, 192, 221, 237, 87, 105, 81, 101, 113, 481, 175, 149, 69, 93, 77, 99, 79, 71, 67, 65, 45, 75, 195, 157, 141, 189, 109, 83, 103, 229, 121, 352, 849, 233, 345, 297, 137, 187, 309, 281, 379, 155, 199, 159

Offset: 1

N. J. A. Sloane, Jan 27 2013

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..7328

Cf. A114183, A189419, A221716, A213220.

# A114181
M:=10000; M2:=1000;
s1:={1}; v0:=[1]; v1:=[1]; v2:=[]; vi:=Array(1..M2);
t1:=1; r1:=1; vi[1]:=1;
for n from 2 to M do
t2:=floor(sqrt(t1));
if t2 in s1 then
   v0:=[op(v0),2*t1]; s1:={op(s1),2*t1}; r1:=r1+1; t1:=2*t1;
   if t1<=M2 then vi[t1]:=n; fi;
else
   v0:=[op(v0),t2]; s1:={op(s1),t2}; v1:=[op(v1),t2]; v2:=[op(v2),r1]; r1:=1; t1:=t2;
   if t1<=M2 then vi[t1]:=n; fi;
fi;
od:
# A114183:
[seq(v0[i],i=1..nops(v0))];
# A221715:
[seq(v1[i],i=1..nops(v1))];
# A221716:
[seq(v2[i],i=1..nops(v2))];
# A189419:
[seq(vi[i],i=1..M2)];

A221716 Length of n-th doubling run in A114183.

Original entry on oeis.org

6, 2, 6, 8, 3, 5, 3, 7, 6, 5, 5, 6, 5, 8, 6, 6, 7, 5, 6, 4, 8, 6, 9, 6, 6, 7, 7, 8, 7, 10, 7, 6, 7, 5, 9, 6, 12, 7, 9, 9, 9, 6, 8, 7, 8, 8, 12, 7, 8, 6, 8, 7, 8, 7, 7, 7, 7, 6, 8, 10, 8, 8, 9, 7, 7, 8, 10, 7, 11, 12, 7, 10, 9, 7, 9, 10, 9, 10, 7, 9, 8, 9, 11, 11, 5, 8, 9, 8, 11, 10, 6, 8, 8, 6, 10, 7, 8, 9, 12, 8, 9, 12, 6, 8, 8, 11, 8, 9, 9, 8, 10, 9, 12

Offset: 1

N. J. A. Sloane, Jan 27 2013

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..7327

Cf. A114183, A189419, A221715, A213220.

Maple
```
See A221715.
```

A213220 Positions in A114183 where doubling runs begin.

Original entry on oeis.org

1, 7, 9, 15, 23, 26, 31, 34, 41, 47, 52, 57, 63, 68, 76, 82, 88, 95, 100, 106, 110, 118, 124, 133, 139, 145, 152, 159, 167, 174, 184, 191, 197, 204, 209, 218, 224, 236, 243, 252, 261, 270, 276, 284, 291, 299, 307, 319, 326, 334, 340, 348, 355, 363, 370, 377, 384, 391, 397, 405, 415, 423, 431, 440, 447, 454, 462, 472

Offset: 1

N. J. A. Sloane, Mar 02 2013

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..7328

Cf. A114183, A221715, A221716 (first differences).

A189419 Inverse permutation to A114183.

Original entry on oeis.org

1, 2, 9, 3, 7, 10, 34, 4, 15, 8, 26, 11, 31, 35, 110, 5, 57, 16, 68, 52, 41, 27, 63, 12, 47, 32, 100, 36, 106, 111, 209, 6, 23, 58, 88, 17, 124, 69, 82, 53, 145, 42, 118, 28, 397, 64, 95, 13, 76, 48, 152, 33, 224

Offset: 1

Franklin T. Adams-Watters, Apr 22 2011

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..3660
Index entries for sequences that are permutations of the natural numbers

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a189419 = (+ 1) . fromJust . (`elemIndex` a114183_list)
-- Reinhard Zumkeller, Mar 05 2013

A222193 Records in A114183.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 48, 96, 144, 288, 576, 1152, 1216, 2432, 4736, 9472, 18688, 37376, 54272, 108544, 115712, 231424, 360448, 720896, 1384448, 2768896, 5537792, 6889472, 13778944, 27557888, 33177600, 66355200, 79495168, 158990336

Offset: 1

N. J. A. Sloane, Feb 13 2013

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..48

Cf. A114183, A222194.

A222194 Where records occur in A114183.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 13, 14, 19, 20, 21, 22, 74, 75, 131, 132, 182, 183, 234, 235, 317, 318, 500, 501, 996, 997, 998, 1549, 1550, 1551, 2347, 2348, 4453, 4454, 4504, 4505, 4506, 4507, 6111, 15261, 15262, 15263, 15264, 56757, 76252, 80742, 80743, 80744

Offset: 1

N. J. A. Sloane, Feb 13 2013

nonn

Cf. A114183, A222193.

A213656 Value of A114183 at end of n-th doubling run.

Original entry on oeis.org

32, 10, 96, 1152, 132, 176, 52, 448, 672, 400, 320, 544, 368, 2432, 1568, 1248, 2240, 752, 864, 232, 1920, 1376, 9472, 3104, 1760, 2624, 3264, 7296, 5440, 37376, 12352, 3552, 3776, 976, 7936, 2848, 108544, 21056, 37120, 49152, 56576, 7584, 11136, 6720, 10368, 12928, 231424, 30784, 22400, 4768, 8832, 5952, 9856

Offset: 1

N. J. A. Sloane, Mar 03 2013

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..7327

Cf. A114183, A221715, A221716, A213220.

A222802 When A114183 decreases in value for the n-th time, dropping to k (say), a(n) is the number of steps earlier that floor(k/2) appeared in A114183.

Original entry on oeis.org

5, 8, 12, 18, 19, 21, 25, 33, 36, 44, 53, 37, 53, 64, 14, 31, 32, 69, 71, 76, 77, 108, 120, 39, 93, 105, 123, 125, 157, 170, 52, 91, 93, 99, 190, 192, 89, 225, 238, 121, 72, 158, 251, 238, 251, 270, 205, 50, 209, 282, 284, 286, 287, 288, 289, 361, 385, 370, 281, 282, 340, 342, 344, 346, 309, 310, 312, 367, 460, 275

Offset: 1

N. J. A. Sloane, Mar 08 2013

nonn

The fact that, when a number k occurs in A114183, floor(k/2) has already appeared, is a key step in the proof that A114183 is a permutation of the natural numbers. This fact is obvious if k is the result of a doubling step. The present sequence is an attempt to gain insight into why it is true when k occurs at a square root step.

			The first 50 terms of A114183 are:
1, 2, 4, 8, 16, 32, 5, 10, 3, 6, 12, 24, 48, 96, 9, 18, 36, 72, 144, 288, 576, 1152, 33, 66, 132, 11, 22, 44, 88, 176, 13, 26, 52, 7, 14, 28, 56, 112, 224, 448, 21, 42, 84, 168, 336, 672, 25, 50, 100, 200.
The sequence decreases from 32 to 5, from 10 to 3, from 96 to 9, and so on.
The values of k are therefore 5, 3, 9, 33, 11, 13, 7, 21, 25, ...
and the corresponding values of floor(k/2) are 2, 1, 4, 16, 5, 6, 3, 10, 12, ...
Since 2 appeared in A114183 5 steps before 5, a(1) = 5,
since 1 appeared 8 steps before 3, a(2) = 8,
since 4 appeared 12 steps before 9, a(3) = 12, and so on.

N. J. A. Sloane, Table of n, a(n) for n = 1..7322

Cf. A114183, A221715, A221716, A213220.

A221718 Floor(sqrt(3*2^n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 19, 27, 39, 55, 78, 110, 156, 221, 313, 443, 627, 886, 1254, 1773, 2508, 3547, 5016, 7094, 10033, 14188, 20066, 28377, 40132, 56755, 80264, 113511, 160529, 227023, 321059, 454046, 642119, 908093, 1284238, 1816186, 2568476, 3632373, 5136952, 7264747, 10273904, 14529495, 20547809, 29058990, 41095618, 58117981

Offset: 0

N. J. A. Sloane, Jan 30 2013

nonn

Theorem 3 of Dubickas implies that infinitely many terms of this sequence are divisible by 2 or 3 (and hence infinitely many composites). - Charles R Greathouse IV, Feb 04 2016

Artūras Dubickas, Prime and composite integers close to powers of a number, Monatsh. Math. 158:3 (2009), pp. 271-284.

N. J. A. Sloane, Table of n, a(n) for n = 0..200

Cf. A007283, A114183.

Floor[Sqrt[3*2^Range[0,50]]] (* Harvey P. Dale, Feb 03 2025 *)

a(n)=sqrtint(3<Charles R Greathouse IV, Feb 04 2016

A221942 a(n) = floor(sqrt(5*2^n)).

Original entry on oeis.org

2, 3, 4, 6, 8, 12, 17, 25, 35, 50, 71, 101, 143, 202, 286, 404, 572, 809, 1144, 1619, 2289, 3238, 4579, 6476, 9158, 12952, 18317, 25905, 36635, 51810, 73271, 103621, 146542, 207243, 293085, 414486, 586171, 828972, 1172343, 1657944, 2344687, 3315888, 4689374, 6631776, 9378748, 13263553, 18757497, 26527107

Offset: 0

N. J. A. Sloane, Feb 01 2013

nonn

Theorem 3 of Dubickas implies that infinitely many terms of this sequence are divisible by 2 or 3 (and hence infinitely many composites). - Charles R Greathouse IV, Feb 04 2016

Artūras Dubickas, Prime and composite integers close to powers of a number, Monatsh. Math. 158:3 (2009), pp. 271-284.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A017910, A017913, A017919, A114183, A221718, A221942-A221946.

Floor[Sqrt[5*2^Range[0,50]]] (* Harvey P. Dale, Mar 09 2025 *)

a(n)=sqrtint(5*2^n) \\ Charles R Greathouse IV, Apr 18 2013

cp's OEIS Frontend

A221715 Start of n-th doubling run in A114183.

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Maple

A221716 Length of n-th doubling run in A114183.

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A213220 Positions in A114183 where doubling runs begin.

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A189419 Inverse permutation to A114183.

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Haskell

A222193 Records in A114183.

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A222194 Where records occur in A114183.

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A213656 Value of A114183 at end of n-th doubling run.

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A222802 When A114183 decreases in value for the n-th time, dropping to k (say), a(n) is the number of steps earlier that floor(k/2) appeared in A114183.

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A221718 Floor(sqrt(3*2^n)).

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Mathematica

PARI

A221942 a(n) = floor(sqrt(5*2^n)).

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PARI