A102370 -id:A102370 - cp's OEIS frontend

A102371 Numbers missing from A102370.

1, 2, 7, 12, 29, 62, 123, 248, 505, 1018, 2047, 4084, 8181, 16374, 32755, 65520, 131057, 262130, 524279, 1048572, 2097133, 4194286, 8388587, 16777192, 33554409, 67108842, 134217711, 268435428, 536870885

Offset: 1

Philippe Deléham, Feb 13 2005

Indices of negative numbers in A103122.
Write numbers in binary under each other; start at 2^k, read in upward direction with the first bit omitted and convert to decimal:
. . . . . . . . . . 0
. . . . . . . . . . 1
.. . . . . . . . . 10 < -- Starting here, the upward diagonal (first bit omitted) reads 1 -> 1
.. . . . . . . . . 11
. . . . . . . . . 100 < -- Starting here, the upward diagonal (first bit omitted) reads 10 -> 2
. . . . . . . . . 101
. . . . . . . . . 110
. . . . . . . . . 111
.. . . . . . . . 1000 < -- Starting here, the upward diagonal (first bit omitted) reads 111 -> 7
. . . . . . . . .1001
Thus a(n) = A102370(2^n - n) - 2^n.
Do we have a(n) = 2^n-1-A105033(n-1)? - David A. Corneth, May 07 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370, A103530, A103581, A103582, A103583, A105033.

a102371 n = a102371_list !! (n-1)
a102371_list = map (a105027 . toInteger) $ tail a000225_list
-- Reinhard Zumkeller, Jul 21 2012

A102371:= proc (n) local t1, l; t1 := -n; for l to n do if `mod`(n-l,2^l) = 0 then t1 := t1+2^l end if end do; t1 end proc;

a=1
for n in range(2,66):
    print(a, end=",")
    a ^= a+n
# Alex Ratushnyak, Apr 21 2012

a(n) = -n + Sum_{ k >= 1, k == n mod 2^k } 2^k. - N. J. A. Sloane and David Applegate, Mar 22 2005. E.g. a(5) = -5 + 2^1 + 2^5 = 29.
a(2^k + k) -a(k) = 2^(2^k + k) - 2^k, with k>= 1.
a(1)=1, for n>1, a(n) = a(n-1) XOR (a(n-1) + n), where XOR is the bitwise exclusive-or operator. - Alex Ratushnyak, Apr 21 2012
a(n) = A105027(A000225(n)). - Reinhard Zumkeller, Jul 21 2012

More terms from Benoit Cloitre, Mar 20 2005
a(16)-a(22) from Robert G. Wilson v, Mar 21 2005
a(15)-a(29) from David Applegate, Mar 22 2005

A103192 Trajectory of 1 under repeated application of the function n -> A102370(n).

Original entry on oeis.org

1, 3, 5, 15, 17, 19, 21, 31, 33, 35, 37, 47, 49, 51, 53, 63, 65, 67, 69, 79, 81, 83, 85, 95, 97, 99, 101, 111, 113, 115, 117, 127, 129, 131, 133, 143, 145, 147, 149, 159, 161, 163, 165, 175, 177, 179, 181, 191, 193, 195, 197, 207, 209, 211, 213, 223, 225, 227, 229, 239, 241

Offset: 1

N. J. A. Sloane, David Applegate, Benoit Cloitre and Philippe Deléham, Mar 25 2005

Agrees with A103127 for the first 511 terms, but then diverges. If a(n) is the present sequence and b(n) is A103127 we have:
.n...a(n)..b(n)..difference
.....................
509, 2033, 2033, 0
510, 2035, 2035, 0
511, 2037, 2037, 0
512, 4095, 2047, 2048
513, 4097, 2049, 2048
514, 4099, 2051, 2048
515, 4101, 2053, 2048
516, 4111, 2063, 2048
.....................
The sequence may be computed as follows (from Philippe Deléham, May 08 2005).
Start with -1, 1. Then add powers of 2 whose exponent n is not in A034797: 1, 3, 11, 2059, 2^2059 + 2059, ... This gives
Step 0: -1, 1
Step 1: add 2^2 = 4, getting 3, 5 and thus -1, 1, 3, 5.
Step 2: add 2^4 = 16, getting 15, 17, 19, 21 and thus -1, 1, 3, 5, 15, 17, 19, 21
Step 3: add 2^5 = 32, getting 31, 33, 35, 37, 47, 49, 51, 53 and thus -1, 1, 3, 5, 15, 17, 19, 21, 31, 33, 35, 37, 47, 49, 51, 53, etc.
The jump 2037 --> 4095 for n = 510 -> 511 is explained by the fact that we pass directly from 2^10 to 2^12 since 11 belongs to A034797.
The trajectories of 2 (A103747) and 7 (A103621) may surely be obtained in a similar way.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences which agree for a long time but are different

a103192 n = a103192_list !! (n-1)
a103192_list = iterate (fromInteger . a102370) 1
-- Reinhard Zumkeller, Jul 21 2012

A103747 Trajectory of 2 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 254, 258, 262, 266, 270, 274, 278, 282, 286, 290, 294, 298, 302, 306, 310, 314, 382, 386, 390, 394, 398, 402, 406, 410, 414, 418, 422

Offset: 1

Benoit Cloitre and David Applegate, Mar 25 2005

Although it initially appears that a(n)-8n is the 16-periodic sequence {-2,-6,-10,-14,-18,-22,-26,-30,-34,-38,-42,-46,-50,-54,6,2}, this pattern eventually breaks down. However, the first divergence occurs beyond the first 400 million terms.

(a(n)) agrees with the 16-periodic sequence up to a(2^67-1) = 2^70 - 70, but then diverges with a(2^67) = 2^71 - 2. - Charlie Neder, Feb 07 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps], preprint, 2005.
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences which agree for a long time but are different

Cf. A102370, A132417.

Trajectories of other numbers: A103192 (1), A103621 (7), A158953 (12), A159887 (29).

a103747 n = a103747_list !! (n-1)
a103747_list = iterate (fromInteger . a102370) 2
-- Reinhard Zumkeller, Jul 21 2012

Edited by Peter Munn, Jan 13 2024

A103621 Trajectory of 7 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

7, 9, 11, 13, 23, 25, 27, 61, 71, 73, 75, 77, 87, 89, 91, 125, 135, 137, 139, 141, 151, 153, 155, 189, 199, 201, 203, 205, 215, 217, 219, 253, 263, 265, 267, 269, 279, 281, 283, 317, 327, 329, 331, 333, 343, 345, 347, 381, 391, 393, 395, 397, 407, 409, 411, 445

Offset: 1

Philippe Deléham, Mar 31 2005

Initially, first differences are 8-periodic: 2,2,2,10,2,2,34,10. [Unsigned comment made accurate by Peter Munn, Jan 13 2024]

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370.

Trajectories of other numbers A103192 (1), A103747 (2), A158953 (12), A159887 (29).

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[ Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s + n]; NestList[f, 7, 55] (* Robert G. Wilson v, Mar 30 2005 *)

Conjectures from Chai Wah Wu, Feb 01 2018: (Start)
a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.
G.f.: x*(3*x^8 + 34*x^7 + 2*x^6 + 2*x^5 + 10*x^4 + 2*x^3 + 2*x^2 + 2*x + 7)/(x^9 - x^8 - x + 1). (End)
The above conjectures are incompatible with A102370(2^37-37) = 2^38-3. - Peter Munn, Jan 13 2024

More terms from Robert G. Wilson v, Mar 30 2005

A104235 Numbers n such that A102370(n) = n.

Original entry on oeis.org

0, 4, 8, 16, 20, 24, 32, 36, 40, 48, 52, 56, 64, 68, 72, 80, 84, 88, 96, 100, 104, 112, 116, 120, 128, 132, 136, 144, 148, 152, 160, 164, 168, 176, 180, 184, 192, 196, 200, 208, 212, 216, 224, 228, 232, 240, 244, 256, 260, 264, 272, 276, 280, 288, 292, 296, 304, 308, 312

Offset: 1

N. J. A. Sloane, Apr 02 2005

See A103543 and A103584 for much more about this sequence.

Indices of the 0 values in A103863.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370, A103543, A103584. Dividing by 4 gives A104401.

Cf. A105271, A214414.

a104235 n = a104235_list !! (n-1)
a104235_list = [x | x <- [0..], a102370 x == toInteger x]
-- Reinhard Zumkeller, Jul 21 2012

A103542 Binary equivalents of A102370.

Original entry on oeis.org

0, 11, 110, 101, 100, 1111, 1010, 1001, 1000, 1011, 1110, 1101, 11100, 10111, 10010, 10001, 10000, 10011, 10110, 10101, 10100, 11111, 11010, 11001, 11000, 11011, 11110, 111101, 101100, 100111, 100010, 100001, 100000, 100011, 100110, 100101

Offset: 0

N. J. A. Sloane, Mar 23 2005

The number of 1's in the n-th term appears to match A089400. - Benoit Cloitre, Mar 24 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
D. Applegate et al., Sloping Binary Number, A New Sequence Related to the Binary Numbers, arXiv:math/0505295 [math.NT], 2005.

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Table[ FromDigits[ IntegerDigits[f[n] + n, 2]], {n, 0, 35}] (* Robert G. Wilson v, Mar 23 2005 *)

def a(n): return '0' if n<1 else bin(sum([(n + k)&(2**k) for k in range(len(bin(n)[2:]) + 1)]))[2:] # Indranil Ghosh, May 03 2017

More terms from Robert G. Wilson v and Benoit Cloitre, Mar 23 2005

A103122 Define a 1-1 correspondence between the integers Z and the nonnegative integers N by f(n) = A102370(n) if n >= 0, f(n) = A102371(-n) if n < 0; sequence gives a(n) = f^(-1)(n) for n >= 0.

Original entry on oeis.org

0, -1, -2, 1, 4, 3, 2, -3, 8, 7, 6, 9, -4, 11, 10, 5, 16, 15, 14, 17, 20, 19, 18, 13, 24, 23, 22, 25, 12, -5, 26, 21, 32, 31, 30, 33, 36, 35, 34, 29, 40, 39, 38, 41, 28, 43, 42, 37, 48, 47, 46, 49, 52, 51, 50, 45, 56, 55, 54, 57, 44, 27, -6, 53, 64, 63, 62, 65, 68

Offset: 0

N. J. A. Sloane, Mar 24 2005

A 1-1 map from the nonnegative integers to all integers.

Simply stated: a(n) = index of n in A102370 if it is a member, else minus its index in the complement A102371. - M. F. Hasler, Apr 14 2022

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

A103122(n)=if(n<0,0,s=-n;while(abs(if(sign(s)+1,2^s-1/2-1/2*sum(k=0,s,(-1)^floor((s+k)/2^k)*2^k),2^(-s-1)-1/2+1/2*sum(k=0,-s-1,(-1)^floor((-s-1-k)/2^k)*2^k))-n)>0,s++);s) \\ Benoit Cloitre, Mar 29 2005

More terms from Benoit Cloitre, Mar 29 2005

A103529 Values of A102370 which are >= a new power of 2.

Original entry on oeis.org

0, 3, 6, 15, 28, 61, 126, 251, 504, 1017, 2042, 4095, 8180, 16373, 32758, 65523, 131056, 262129, 524274, 1048567, 2097148, 4194285, 8388590, 16777195, 33554408, 67108841, 134217706, 268435439, 536870884, 1073741797, 2147483622

Offset: 1

N. J. A. Sloane and David Applegate, Mar 22 2005

			The initial values of A102370 are 0*, 3*, 6*, 5, 4, 15*, 10, 9, 8, 11, 14, 13, 28*, 23, ... and the starred terms are those which exceed the next power of 2. Their indices (except for the zero term) are given by A000325.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..62
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370, A103530.

a=3
print(0, end=',')
for i in range(2,55):
    print(a, end= ',')
    a ^= a+i
# Alex Ratushnyak, Apr 21 2012

a(n) = 2^(n-1) - (n-1) + Sum_{ k >= 1, k == n-1 mod 2^k } 2^k.
a(n+1) = 2^n + A102371(n) for n>=1. a(n) = 2^n - A103530(n). - Philippe Deléham, Mar 30 2005
a(0)=0, a(1)=3, for n>1, a(n)= a(n-1) XOR (a(n-1)+n), where XOR is the bitwise exclusive-or operator. - Alex Ratushnyak, Apr 21 2012

A158953 Trajectory of 12 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

12, 28, 44, 60, 76, 92, 108, 124, 140, 156, 172, 188, 204, 220, 236, 252, 268, 284, 300, 316, 332, 348, 364, 380, 396, 412, 428, 444, 460, 476, 492, 508, 524, 540, 556, 572, 588, 604, 620, 636, 652, 668, 684, 700, 716, 732, 748, 764, 780, 796, 812, 828, 844

Offset: 1

Philippe Deléham, Apr 01 2009

Coincides with A098502 for at least 1400 terms. - R. J. Mathar, Apr 16 2009

Agrees with A098502 for the first 65535 terms. A098502(65535) = a(65535) = 1048556 = 2^20 - 20. A098502(65536) = 1048572 = 2^20 - 4; a(65536) = 2097148 = 2^21 - 4. - Philippe Deléham, Jan 05 2023

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [pdf, ps].
Index entries for sequences which agree for a long time but are different

Cf. A098502, A102370.

Trajectories of other numbers: A103192 (1), A103747 (2), A103621 (7), A159887 (29).

A159887 Trajectory of 29 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

29, 39, 41, 43, 45, 55, 57, 59, 93, 103, 105, 107, 109, 119, 121, 251, 285, 295, 297, 299, 301, 311, 313, 315, 349, 359, 361, 363, 365, 375, 377, 507, 541, 551, 553, 555, 557, 567, 569, 571, 605, 615, 617, 619, 621, 631, 633, 763, 797, 807, 809, 811, 813, 823, 825

Offset: 1

Philippe Deléham, Apr 25 2009

Not the same as A159888: see the comments in A159888.

The divergence from A159888 follows from Theorem 3.1 in the Applegate, Cloitre, Deléham and Sloane link: in general, the first differences of an A102370 trajectory cannot be a cycle. - Peter Munn, Jan 14 2024

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences which agree for a long time but are different

Cf. A102370, A159888.

Trajectories of other numbers: A103192 (1), A103747 (2), A103621 (7), A158953 (12).

Missing term 617 inserted by Georg Fischer, Nov 28 2023

cp's OEIS Frontend

A102371 Numbers missing from A102370.

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A103192 Trajectory of 1 under repeated application of the function n -> A102370(n).

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A103747 Trajectory of 2 under repeated application of the map n -> A102370(n).

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A103621 Trajectory of 7 under repeated application of the map n -> A102370(n).

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A104235 Numbers n such that A102370(n) = n.

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A103542 Binary equivalents of A102370.

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A103122 Define a 1-1 correspondence between the integers Z and the nonnegative integers N by f(n) = A102370(n) if n >= 0, f(n) = A102371(-n) if n < 0; sequence gives a(n) = f^(-1)(n) for n >= 0.

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A103529 Values of A102370 which are >= a new power of 2.

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