A110287 -id:A110287 - cp's OEIS frontend

A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048

Offset: 0

Reinhard Zumkeller, Feb 28 2010

Essentially the same as A048645. - T. D. Noe, Mar 28 2011

			Triangle begins as:
     2;
     3,    4;
     5,    6,    8;
     9,   10,   12,   16;
    17,   18,   20,   24,   32;
    33,   34,   36,   40,   48,   64;
    65,   66,   68,   72,   80,   96,  128;
   129,  130,  132,  136,  144,  160,  192,  256;
   257,  258,  260,  264,  272,  288,  320,  384,  512;
   513,  514,  516,  520,  528,  544,  576,  640,  768, 1024;
  1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;

T. D. Noe, Rows n = 0..100 of triangle, flattened
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).

Cf. A048645, A118413, A118416.

Cf. also A087112, A370121.

[2^n + 2^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 07 2021

Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* T. D. Noe, Jun 18 2013 *)

A173786(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)); }; \\ Antti Karttunen, Feb 29 2024, after David A. Corneth's PARI-program in A048645

from math import isqrt, comb
def A173786(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return (1<Chai Wah Wu, Jun 20 2025

flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 07 2021

1 <= A000120(T(n,k)) <= 2.
For n>0, 0<=kA048645(n+1,k+2) and T(n,n) = A048645(n+2,1).
Row sums give A006589(n).
Central terms give A161168(n).
T(2*n+1,n) = A007582(n+1).
T(2*n+1,n+1) = A028403(n+1).
T(n,k) = A140513(n,k) - A173787(n,k), 0<=k<=n.
T(n,k) = A059268(n+1,k+1) + A173787(n,k), 0T(n,k) * A173787(n,k) = A173787(2*n,2*k), 0<=k<=n.
T(n,0) = A000051(n).
T(n,1) = A052548(n) for n>0.
T(n,2) = A140504(n) for n>1.
T(n,3) = A175161(n-3) for n>2.
T(n,4) = A175162(n-4) for n>3.
T(n,5) = A175163(n-5) for n>4.
T(n,n-4) = A110287(n-4) for n>3.
T(n,n-3) = A005010(n-3) for n>2.
T(n,n-2) = A020714(n-2) for n>1.
T(n,n-1) = A007283(n-1) for n>0.
T(n,n) = 2*A000079(n).

Typo in first comment line fixed by Reinhard Zumkeller, Mar 07 2010

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

Original entry on oeis.org

6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 1

Enoch Haga, Jan 24 2004

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing digit "1".

The monotonic increasing value of successive product of digits strongly suggests that in successive n the digit 1 must be present.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences
Carlos Rivera, Puzzle 251, Pointer primes, The Prime Puzzles and Problems Connection.
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Cf. A089823, A091628, A091630, A091631, A091632.
Similar to A003945, A042950, A058764, A087009, A081808.

[3*2^n : n in [1..40]]; // Wesley Ivan Hurt, Jul 17 2020

3*2^Range[1, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

[3*2^n for n in range(1,51)] # G. C. Greubel, Jan 05 2023

a(n) = 3 * 2^n = product of digits of A091628(n).
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 6*2^(n-1).
a(n) = 2*a(n-1), with a(1) = 6.
G.f.: 6*x/(1-2*x). (End)
E.g.f.: 3*(exp(2*x) - 1). - G. C. Greubel, Jan 05 2023

Edited and extended by Ray Chandler, Feb 07 2004

A334101 Numbers of the form q*(2^k), where q is one of the Fermat primes and k >= 0; Numbers n for which A329697(n) == 1.

Original entry on oeis.org

3, 5, 6, 10, 12, 17, 20, 24, 34, 40, 48, 68, 80, 96, 136, 160, 192, 257, 272, 320, 384, 514, 544, 640, 768, 1028, 1088, 1280, 1536, 2056, 2176, 2560, 3072, 4112, 4352, 5120, 6144, 8224, 8704, 10240, 12288, 16448, 17408, 20480, 24576, 32896, 34816, 40960, 49152, 65537, 65792, 69632, 81920, 98304, 131074, 131584, 139264

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Numbers k that themselves are not powers of two, but for which A171462(k) = k-A052126(k) is [a power of 2].
Numbers k such that A000265(k) is in A019434.
Squares of these numbers can be found (as a subset) in A334102, and the cubes (as a subset) in A334103.

Row 1 of A334100.
Cf. A000120, A000265, A052126, A171462, A209229, A329697, A334102, A334103.
Cf. A019434 (primes present), A007283, A020714, A110287 (other subsequences).
Subsequence of A018900.

A000265(n) = (n>>valuation(n,2));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2,n-1));
isA334101(n) = isA019434(A000265(n));

A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
A209229(n) = (n && !bitand(n,n-1));
isA334101(n) = ((!A209229(n))&&A209229(n-A052126(n)));

For all n, A000120(a(n)) = 2.

A110288 a(n) = 19*2^n.

Original entry on oeis.org

19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 19456, 38912, 77824, 155648, 311296, 622592, 1245184, 2490368, 4980736, 9961472, 19922944, 39845888, 79691776, 159383552, 318767104, 637534208, 1275068416, 2550136832, 5100273664, 10200547328, 20401094656

Offset: 0

Alexandre Wajnberg, Sep 07 2005

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).

19 times powers of 2. - Omar E. Pol, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..235
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), this sequence (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).

[19*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

19*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,19,30] (* Harvey P. Dale, May 11 2018 *)

[19*2^n for n in range(41)] # G. C. Greubel, Jan 04 2023

G.f.: 19/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = A000079(n)*19. - Omar E. Pol, Dec 17 2008
E.g.f.: 19*exp(2*x). - G. C. Greubel, Jan 04 2023

Edited by Omar E. Pol, Dec 16 2008

A198275 a(n) = 17*2^n - 1.

Original entry on oeis.org

16, 33, 67, 135, 271, 543, 1087, 2175, 4351, 8703, 17407, 34815, 69631, 139263, 278527, 557055, 1114111, 2228223, 4456447, 8912895, 17825791, 35651583, 71303167, 142606335, 285212671, 570425343, 1140850687, 2281701375, 4563402751, 9126805503, 18253611007

Offset: 0

Jeremy Gardiner, Oct 23 2011

			a(2) = 17*2^2-1 = 67.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3, 14.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A086224, A110287.

for j = 0 to 30 : print str$((17*2^j)-1)+", "; : next j

[17*2^n-1: n in [0..30]]; // Vincenzo Librandi, Oct 28 2011

A198275:=n->17*2^n-1; seq(A198275(n), n=0..30); # Wesley Ivan Hurt, Jun 11 2014

17*2^Range[0,30]-1 (* or *) LinearRecurrence[{3,-2},{16,33},40] (* Harvey P. Dale, Dec 03 2011 *)

a(n+1) = 2*a(n) + 1.
G.f.: ( 16-15*x ) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Oct 25 2011
a(0)=16, a(1)=33, a(n)=3*a(n-1)-2*a(n-2). - Harvey P. Dale, Dec 03 2011
a(n) + a(n-1)^2 = (a(n-1)+1)^2. - Vincenzo Librandi, Jun 11 2014

A239303 Triangle of compressed square roots of Gray code * bit-reversal permutation.

Original entry on oeis.org

1, 3, 1, 6, 1, 5, 6, 9, 1, 10, 12, 18, 1, 17, 10, 12, 18, 33, 1, 34, 20, 24, 36, 66, 1, 65, 34, 20, 24, 36, 66, 129, 1, 130, 68, 40, 48, 72, 132, 258, 1, 257, 130, 68, 40, 48, 72, 132, 258, 513, 1, 514, 260, 136, 80

Offset: 1

Tilman Piesk, Mar 14 2014

The permutation that turns a natural ordered into a sequency ordered Walsh matrix of size 2^n is the product of the Gray code permutation A003188(0..2^n-1) and the bit-reversal permutation A030109(n,0..2^n-1).
(This permutation of 2^n elements can be represented by the compression vector [2^(n-1), 3*[2^(n-2)..4,2,1]] with n elements.)
This triangle shows the compression vectors of the unique square roots of these permutations, which correspond to symmetric binary matrices with 2n-1 ones.
(These n X n matrices correspond to graphs that can be described by permutations of n elements, which are shown in A239304.)
Rows of the square array:
T(1,n) = 1,3,6,6,12,12,24,24,48,48,96,96,192,192,384,384,... (compare A003945)
T(2,n) = 1,1,9,18,18,36,36,72,72,144,144,288,288,576,576,... (compare A005010)
Columns of the square array:
T(m,1) = 1,1,5,10,10,20,20,40,40,80,80,160,160,320,320,... (compare A146523)
T(m,2) = 3,1,1,17,34,34,68,68,136,136,272,272,544,544,... (compare A110287)

			Triangular array begins:
   1
   3   1
   6   1   5
   6   9   1  10
  12  18   1  17  10
  12  18  33   1  34  20
Square array begins:
   1   3   6   6  12  12
   1   1   9  18  18  36
   5   1   1  33  66  66
  10  17   1   1 129 258
  10  34  65   1   1 513
  20  34 130 257   1   1
The Walsh permutation wp(8,12,6,3) = (0,8,12,4, 6,14,10,2, 3,11,15,7, 5,13,9,1) permutes the natural ordered into the sequency ordered Walsh matrix of size 2^4.
Its square root is wp(6,9,1,10) = (0,6,9,15, 1,7,8,14, 10,12,3,5, 11,13,2,4).
So row 4 of the triangular array is (6,9,1,10).

Tilman Piesk, First 140 rows of the triangle, flattened
Tilman Piesk, Sequency ordered Walsh matrix (Wikiversity)
Tilman Piesk, Calculation in MATLAB

Cf. A239304, A003188, A030109.

A159026 a(0)=89; a(n) = a(n-1) + floor(sqrt(a(n-1))), n > 0.

Original entry on oeis.org

89, 98, 107, 117, 127, 138, 149, 161, 173, 186, 199, 213, 227, 242, 257, 273, 289, 306, 323, 340, 358, 376, 395, 414, 434, 454, 475, 496, 518, 540, 563, 586, 610, 634, 659, 684, 710, 736, 763, 790, 818, 846, 875, 904, 934, 964, 995, 1026, 1058, 1090, 1123, 1156, 1190

Offset: 0

Philippe Deléham, Apr 02 2009

nonn

Row 8 in square array A159016.

This sequence contains infinitely many squares. - Philippe Deléham, Apr 04 2009

Cf. A028392, A110287.

NestList[#+Floor[Sqrt[#]]&,89,60] (* Harvey P. Dale, Feb 26 2012 *)

More terms from Vincenzo Librandi, Apr 10 2009

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cp's OEIS Frontend

A005009 a(n) = 7*2^n.

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A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

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A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

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A334101 Numbers of the form q*(2^k), where q is one of the Fermat primes and k >= 0; Numbers n for which A329697(n) == 1.

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A110288 a(n) = 19*2^n.

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A198275 a(n) = 17*2^n - 1.

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