A029744 -id:A029744 - cp's OEIS frontend

A163978 a(n) = 2*a(n-2) for n > 2; a(1) = 3, a(2) = 4.

3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728

Offset: 1

Klaus Brockhaus, Aug 07 2009

Interleaving of A007283 and A000079 without initial terms 1 and 2.
Equals A029744 without first two terms. Agrees with A145751 for all terms listed there (up to 65536).
Binomial transform is A078057 without initial 1, second binomial transform is A048580, third binomial transform is A163606, fourth binomial transform is A163604, fifth binomial transform is A163605.
a(n) is the number of vertices of the (n-1)-iterated line digraph L^{n-1}(G) of the digraph G(=L^0(G)) with vertices u,v,w and arcs u->v, v->u, v->w, w->v. - Miquel A. Fiol, Jun 08 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Miquel A. Fiol, J. L. A. Yebra, and I. Alegre, Line digraph iterations and the (d,k) digraph problem, IEEE Trans. Comput. C-33(5) (1984), 400-403.
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A027383, A029744, A048580, A052955, A063759, A078057, A090989, A145751, A163604, A163605, A163606.

[ n le 2 select n+2 else 2*Self(n-2): n in [1..41] ];

LinearRecurrence[{0,2}, {3,4}, 52] (* or *) Table[(1/2)*(5-(-1)^n )*2^((2*n-1+(-1)^n)/4), {n,50}] (* G. C. Greubel, Aug 24 2017 *)

my(x='x+O('x^50)); Vec(x*(3+4*x)/(1-2*x^2)) \\ G. C. Greubel, Aug 24 2017

[(2+(n%2))*2^((n-(n%2))//2) for n in range(1,41)] # G. C. Greubel, Jun 13 2024

a(n) = A027383(n-1) + 2.
a(n) = A052955(n) + 1 for n >= 1.
a(n) = (1/2)*(5 - (-1)^n)*2^((2*n - 1 + (-1)^n)/4).
G.f.: x*(3+4*x)/(1-2*x^2).
a(n) = A090989(n-1).
E.g.f.: (1/2)*(4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) - 4). - G. C. Greubel, Aug 24 2017
a(n) = A063759(n), n >= 1. - R. J. Mathar, Jan 25 2023

A166278 Square array A(n,k), n,k>=0, read by antidiagonals: A(n,k) is the total element sum of the k-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 6, 10, 10, 5, 0, 1, 8, 19, 20, 15, 6, 0, 1, 12, 33, 46, 35, 21, 7, 0, 1, 16, 63, 100, 94, 56, 28, 8, 0, 1, 24, 111, 220, 242, 172, 84, 36, 9, 0, 1, 32, 201, 488, 633, 514, 290, 120, 45, 10, 0, 1, 48, 369, 1104, 1643, 1518, 984, 460, 165, 55, 11

Offset: 0

Alois P. Heinz, Oct 10 2009

			A(3,4) = 33, because f([1,1,1]) = [1,2,3], (f^2)([1,1,1]) = [3,3,4], (f^3)([1,1,1]) = [4,6,9], (f^4)([1,1,1]) = [9,12,12], and 9+12+12 = 33.
Square array A(n,k) begins:
  0,  0,  0,  0,   0,   0, ...
  1,  1,  1,  1,   1,   1, ...
  2,  3,  4,  6,   8,  12, ...
  3,  6, 10, 19,  33,  63, ...
  4, 10, 20, 46, 100, 220, ...
  5, 15, 35, 94, 242, 633, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-3 give: A001477, A000217, A000292, A070893.
Rows n=0-2 give: A000004, A000012, A029744(k+2).
Main diagonal gives A261490.

f:= l-> sort([seq(sort(l, `>`)[i]*i, i=1..nops(l))]):
A:= (n, k)-> add(i, i=(f@@k)([1$n])):
seq(seq(A(n, d-n), n=0..d), d=0..15);

f[L_List] := f[L] = Sort[Reverse[Sort[L]]*Range[Length[L]]];
A[0, ] = 0; A[n, 0] := n; A[n_, k_] := Total[Nest[f, Range[n], k-1]];
Table[A[n, k-n], {k, 0, 15}, {n, 0, k}] // Flatten (* Jean-François Alcover, Jun 07 2016 *)

A185333 Number of binary necklaces of n beads for which a cut exists producing a palindrome.

Original entry on oeis.org

2, 2, 4, 3, 8, 6, 16, 9, 32, 20, 64, 34, 128, 72, 256, 129, 512, 272, 1024, 516, 2048, 1056, 4096, 2050, 8192, 4160, 16384, 8200, 32768, 16512, 65536, 32769, 131072, 65792, 262144, 131088, 524288, 262656, 1048576, 524292, 2097152

Offset: 1

Tony Bartoletti, Feb 20 2011

nonn

For odd n, this corresponds to A029744, necklaces of n beads that are the same when turned over. For even n, consider the necklace "01", which satisfies A029744 but cannot be cut to produce a palindrome.

Hugo Pfoertner and Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A185376, A025480. - Hugo Pfoertner, Jul 30 2011

f[n_] := If[OddQ@ n, 2^(n/2 + 1/2), Block[{k = IntegerExponent[n, 2] - 1}, 2^(n/2 - 1) + 2^((n/2 - 2^k)/(2^(k + 1)))]]; Array[f, 41] (* Robert G. Wilson v, Aug 08 2011 *)

def a(n):
    if n%2: return 2**((n + 1)//2)
    k=bin(n - 1)[2:].count('1') - bin(n)[2:].count('1')
    return 2**(n//2 - 1) + 2**((n//2 - 2**k)//(2**(k + 1)))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 29 2017

A339893 a(n) = A000523(n) - A001222(n); floor(log_2(n)) minus the number of prime divisors of n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 3, 1, 2, 2, 3, 0, 2, 2, 1, 1, 3, 1, 3, 0, 3, 3, 3, 1, 4, 3, 3, 1, 4, 2, 4, 2, 2, 3, 4, 0, 3, 2, 3, 2, 4, 1, 3, 1, 3, 3, 4, 1, 4, 3, 2, 0, 4, 3, 5, 3, 4, 3, 5, 1, 5, 4, 3, 3, 4, 3, 5, 1, 2, 4, 5, 2, 4, 4, 4, 2, 5, 2, 4, 3, 4, 4, 4, 0, 5, 3, 3, 2, 5, 3, 5, 2, 3

Offset: 1

Antti Karttunen, Dec 21 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000523, A001222, A029744 (positions of 0's), A339895.

Cf. also A339823, A342657 [= a(A156552(n))].

A339893(n) = (#binary(n) - 1 - bigomega(n));

a(n) = A000523(n) - A001222(n).

a(n) = A339895(A122111(n)).

A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).

Original entry on oeis.org

4, 9, 16, 36, 64, 144, 256, 576, 1024, 2304, 4096, 9216, 16384, 36864, 65536, 147456, 262144, 589824, 1048576, 2359296, 4194304, 9437184, 16777216, 37748736, 67108864, 150994944, 268435456, 603979776, 1073741824, 2415919104, 4294967296, 9663676416, 17179869184

Offset: 1

Karl-Heinz Hofmann, Sep 02 2022

If x is even, y = x + 3; if x is odd, y = x.
Proof for odd x: (2^odd + 2^odd) = 2^(odd + 1) = 2^even --> must be a square.
Proof for even x: 2^even + 2^(even + 3) = 1*(2^even) + (2^even * 2^3) = 1*(2^even) + (2^even * 8) = 1*(2^even) + 8*(2^even) = 9*(2^even); since 9 is a square and 2^even is a square, the multiplication result must be a square too.
And 9 is the only square that can be written as 1 + a power of 2.
Note that a(n) = A272711(n+1) for n=1..23, but beyond it differs more and more.

			2^4 + 2^7 = 144, a square, thus 144 is a term.

Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A029744, A000302, A002063.
Intersection of A000290 and A048645\{1}.
Cf. A272711, A270473 (squares that can be expressed as 3^x + 3^y).
Cf. A220221.

seq(`if`(n::even, 9*2^(n-2), 2^(n+1)),n=1..50); # Robert Israel, Sep 15 2022

Select[Range[2, 2^17]^2, DigitCount[#, 2, 1] <= 2 &] (* Amiram Eldar, Sep 03 2022 *)

a(n) = if (n%2, 2^(n+1), 9*2^(n-2)); \\ Michel Marcus, Sep 15 2022

def A356880(n):
    if n % 2 == 0: return 9*2**(n-2)
    else: return 2**(n+1)

a(n) = A029744(n+1)^2.
a(n) = 9 * 2^(n-2) if n is even (see A002063).
a(n) = 2^(n+1) if n is odd (see A000302).
From Stefano Spezia, Sep 09 2022: (Start)
G.f.: x*(4 + 9*x)/(1 - 4*x^2).
E.g.f.: (9*(cosh(2*x) - 1) + 8*sinh(2*x))/4. (End)

A364956 Numbers k such that A163511(k) is either k itself or its descendant in Doudna-tree, A005940 (or equally, in A163511-tree).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 341887, 393216, 524288, 683774, 786432, 1048576, 1572864, 2097152, 2495625, 3145728, 4194304, 4991250, 6291456

Offset: 1

Antti Karttunen, Sep 02 2023

nonn

Numbers k such that A252464(k) = A364954(k), where A364954(n) is the length of the common prefix in the binary expansions of A156552(n) and A156552(A163511(n)).

			For n = 341887, A156552(n) = 1736, "11011001000" in binary, and A163511(n) = 1830711541, with A156552(A163511(n)) = 444544, "1101100100010000000" in binary, and as the former binary expansion is a prefix of the latter, 341887 is included in this sequence. In this case, 1830711541 = A003961^7(2*341887), where A003961^7 indicates a prime shift by seven steps towards larger primes.
For n = 683774 = 2*341887, A156552(n) = 3473 = "110110010001", and A163511(n) = 3661423082 = 2*1830711541, with A156552(A163511(n)) = 889089, "11011001000100000001", and as the former binary expansion is a prefix of the latter, 683774 is included in this sequence.
For n = 1367548 = 4*341887, A156552(n) = 6947, "1101100100011" in binary, and A163511(n) = 7322846164 = 2*3661423082 with A156552(A163511(n)) = 1778179, "110110010001000000011" in binary, as the former binary expansion is NOT a prefix of the latter, 1367548 is NOT included in this sequence.

Positions of 0's in A364955.
Cf. A005940, A156552, A163511.
Cf. A029744 (subsequence).
Cf. also A364960.
Cf. A252464, A364954.

A377000 Array read by ascending antidiagonals: T(n,k) = number of n-esthetic numbers with k digits.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 8, 6, 1, 6, 9, 12, 13, 8, 1, 7, 11, 16, 21, 21, 12, 1, 8, 13, 20, 29, 36, 34, 16, 1, 9, 15, 24, 37, 52, 63, 55, 24, 1, 10, 17, 28, 45, 68, 94, 108, 89, 32, 1, 11, 19, 32, 53, 84, 126, 169, 189, 144, 48, 1, 12, 21, 36, 61, 100, 158, 232, 305, 324, 233, 64, 1

Offset: 2

Paolo Xausa, Oct 12 2024

A number is n-esthetic if, when written in base n, adjacent digits differ by 1: see De Koninck and Doyon (2009), where T(n,k) is denoted by N_q(r).

			Array begins (cf. De Koninck and Doyon (2009), table on p. 155):
  n\k| 1   2   3   4    5    6    7    8     9    10  ...
  -------------------------------------------------------
   2 | 1,  1,  1,  1,   1,   1,   1,   1,    1,    1, ... = A000012
   3 | 2,  3,  4,  6,   8,  12,  16,  24,   32,   48, ... = A029744 (from n = 2)
   4 | 3,  5,  8, 13,  21,  34,  55,  89,  144,  233, ... = A000045 (from n = 4)
   5 | 4,  7, 12, 21,  36,  63, 108, 189,  324,  567, ... = A228879
   6 | 5,  9, 16, 29,  52,  94, 169, 305,  549,  990, ...
   7 | 6, 11, 20, 37,  68, 126, 232, 430,  792, 1468, ...
   8 | 7, 13, 24, 45,  84, 158, 296, 557, 1045, 1966, ...
   9 | 8, 15, 28, 53, 100, 190, 360, 685, 1300, 2475, ...
  10 | 9, 17, 32, 61, 116, 222, 424, 813, 1556, 2986, ... = A090994
  ...                                               \______ A152086 (main diagonal)

Paolo Xausa, Table of n, a(n) for n = 2..11326 (first 150 antidiagonals, flattened).
Jean-Marie De Koninck and Nicolas Doyon, Esthetic Numbers, Ann. Sci. Math. Québec 33 (2009), No. 2, pp. 155-164.
Giovanni Resta, Esthetic Numbers, Numbers Aplenty, 2013.
Branko J. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33 (arXiv version, arXiv:0704.0750 [math.DG], 2007).

Cf. A000012 (row n = 2), A029744 (row n = 3), A000045 (row n = 4), A228879 (row n = 5), A090994 (row n = 10).
Cf. A102699, A152086 (main diagonal).
Diagonal above the main diagonal appears to be A206603.

A377000[n_, k_] := Round[2^k/(n+1)*Sum[If[m != (n+1)/2, Cos[#]^k*(Cot[#] + Csc[#])^2 & [Pi*m/(n+1)], 0], {m, 1, n, 2}]];
Table[A377000[n-k+1, k], {n, 2, 15}, {k, n-1}]

from itertools import count, islice
from functools import lru_cache
@lru_cache(maxsize=None)
def A377000_N(q,r,i):
    if r==1 and i==0: return 0
    if r==1: return 1
    if q==2: return r+i&1^1
    if i == 0: return A377000_N(q,r-1,1)
    if i == q-1: return A377000_N(q,r-1,q-2)
    return A377000_N(q,r-1,i-1)+A377000_N(q,r-1,i+1)
def A377000_T(n,k): return sum(A377000_N(n,k,i) for i in range(n))
def A377000_gen(): # generator of terms
    for n in count(2):
        for k in range(1,n):
            yield A377000_T(n-k+1,k)
A377000_list = list(islice(A377000_gen(),100)) # Chai Wah Wu, Oct 21 2024

All of the following formulas are taken from De Koninck and Doyon (2009).
T(n,k) = 2^k/(n+1) * Sum_{m=1..n, m odd, m != (n+1)/2} cos(p)^k*(cot(p) + csc(p))^2, where p = Pi*m/(n+1).
T(n,1) = n - 1.
T(2,k) = 1.
T(3,k) = 2^((k+1)/2) if k is odd, 3*2^((k-2)/2) if k is even = A029744(k+1).
T(4,k) = A000045(k+3).
T(5,k) = 4*3^((k-1)/2) if k is odd, 7*3^((k-2)/2) if k is even = A228879(k-1).
Conjectures from Chai Wah Wu, Oct 21 2024: (Start)
Conjecture 1: For even n, T(n,k) is the number of meaningful differential operations of the k-th order on the space R^(n-1).
Conjecture 2: For each n, the row T(n,k) satisfies a linear recurrence. For example:
T(6,k) = T(6,k-1) + 2*T(6,k-2) - T(6,k-3) for k > 3 (A090990).
T(7,k) = 4*T(7,k-2) - 2*T(7,k-4) for k > 4.
T(8,k) = T(8,k-1) + 3*T(8,k-2) - 2*T(8,k-3) - T(8,k-4) for k > 4 (A090992).
T(9,k) = 5*T(9,k-2) - 5*T(9,k-4) for k > 4.
T(10,k) = T(10,k-1) + 4*T(10,k-2) - 3*T(10,k-3) - 3*T(10,k-4) + T(10,k-5) for k > 5.
T(11,k) = 6*T(11,k-2) - 9*T(11,k-4) + 2*T(11,k-6) for k > 6.
T(12,k) = T(12,k-1) + 5*T(12,k-2) - 4*T(12,k-3) - 6*T(12,k-4) + 3*T(12,k-5) + T(12,k-6) for k > 6 (A129638).
...
Note that for even n, Conjecture 1 implies Conjecture 2 due to (Malesevic, 1998).
Conjecture 3: T(n,n-2) = A182555(n-2). (End)

A258209 Numbers k for which A256999(A059893(k)) = k.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 26, 28, 30, 31, 32, 48, 52, 56, 58, 60, 62, 63, 64, 96, 100, 104, 106, 112, 118, 120, 122, 124, 126, 127, 128, 192, 200, 208, 212, 224, 228, 234, 236, 240, 246, 248, 250, 252, 254, 255, 256, 384, 392, 400, 416, 420, 424, 426, 448, 460, 466, 472, 474, 480, 484, 490, 494, 496, 502, 504, 506, 508, 510, 511, 512

Offset: 0

Antti Karttunen, May 31 2015

Indexing starts from zero, because a(0) = 0 is a special case.

These numbers correspond to the maximal (lexicographically largest) representatives selected from each equivalence class of those binary necklaces that stay the same (in the same equivalence class) when flipped over (which thus have a bilateral symmetry, please see the examples). A029744(n) gives the number of terms with n significant bits in their binary representation.

			28 ("11100" in binary) is in sequence, because after removing the most significant bit, the binary string "1100" when reversed, "0011", can then be rotated (two steps in either direction) to give "1100" again and "1100" is the lexicographically largest of these rotations.
114 ("1110010" in binary) is NOT in the sequence, because after removing the most significant bit, the binary string "110010" when reversed, "010011", does not yield "110010" no matter how many steps it is rotated (even though it is the lexicographically largest rotation of its class). Thus although 114 is in A257250 (a supersequence of this sequence), it is not included here.

Antti Karttunen, Table of n, a(n) for n = 0..5117

Cf. A029744, A059893, A256999.
Subsequence of A257250.
Differs from A257250 for the first time at n=31, where a(31) = 118, while A257250(31) = 114.

A265887 T(n,k)=Number of nXk 0..2 arrays with the sum of the absolute differences of each element with its horizontal and vertical neighbors equal to the number of neighbors.

Original entry on oeis.org

3, 4, 4, 6, 12, 6, 8, 16, 16, 8, 12, 32, 68, 32, 12, 16, 64, 128, 128, 64, 16, 24, 128, 384, 664, 384, 128, 24, 32, 256, 1024, 2048, 2048, 1024, 256, 32, 48, 512, 3072, 8192, 13672, 8192, 3072, 512, 48, 64, 1024, 8192, 32768, 65536, 65536, 32768, 8192, 1024, 64, 96

Offset: 1

R. H. Hardin, Dec 17 2015

Table starts
..3....4.....6.......8.......12.........16..........24............32
..4...12....16......32.......64........128.........256...........512
..6...16....68.....128......384.......1024........3072..........8192
..8...32...128.....664.....2048.......8192.......32768........131072
.12...64...384....2048....13672......65536......393216.......2097152
.16..128..1024....8192....65536.....560512.....4194304......33554432
.24..256..3072...32768...393216....4194304....51483264.....536870912
.32..512..8192..131072..2097152...33554432...536870912....8726974464
.48.1024.24576..524288.12582912..268435456..6442450944..137438953472
.64.2048.65536.2097152.67108864.2147483648.68719476736.2199023255552

			Some solutions for n=4 k=4
..2..1..2..1....0..0..1..0....1..2..1..0....1..2..1..2....2..1..2..1
..1..2..1..2....2..2..2..1....0..1..2..1....0..1..0..1....1..2..1..0
..2..1..2..1....1..0..0..2....1..2..1..2....1..2..1..0....0..1..2..1
..1..2..1..0....2..1..0..2....0..1..0..1....2..1..2..1....1..2..1..0

R. H. Hardin, Table of n, a(n) for n = 1..684

Column 1 is A029744(n+2).

Empirical for column k:
k=1: a(n) = 2*a(n-2)
k=2: a(n) = 2*a(n-1) for n>3
k=3: a(n) = 8*a(n-2) for n>5
k=4: a(n) = 4*a(n-1) for n>5
k=5: a(n) = 32*a(n-2) for n>7
k=6: a(n) = 8*a(n-1) for n>7
k=7: a(n) = 128*a(n-2) for n>9

A272711 Square numbers whose binary reversal is also square.

Original entry on oeis.org

1, 4, 9, 16, 36, 64, 144, 256, 576, 1024, 2304, 4096, 9216, 16384, 36864, 65536, 147456, 262144, 589824, 1048576, 2359296, 4194304, 9437184, 16777216, 20457529, 37748736, 67108864, 81830116, 143784081, 150994944, 268435456, 327320464, 331130809, 575136324, 603979776

Offset: 1

Benjamin Przybocki, May 04 2016

The first term in this sequence whose binary reversal is not 1 or 9 is 20457529 (which is a binary palindrome).
The previous comment means that the sequence does not just contain the squares of numbers in A029744. - R. J. Mathar, May 06 2016
If k is a term, then so is 4*k. - Robert Israel, Jun 06 2023

Robert Israel, Table of n, a(n) for n = 1..176

Cf. A000290, A030101, A029983, A229687.

Cf. A061457 (analogous in base 10).

rev:= proc(n) local L,i;
  L:= convert(n,base,2);
  add(L[-i]*2^(i-1),i=1..nops(L))
end proc:
select(n -> issqr(rev(n)), [seq(i^2,i=1..100000)]); # Robert Israel, Jun 06 2023

Select[Range[10^5]^2, IntegerQ@ Sqrt@ FromDigits[ Reverse@ IntegerDigits[#, 2], 2] &] (* Giovanni Resta, May 05 2016 *)

lista(nn) = {for (n=1, nn, if (issquare(subst(Polrev(binary(n^2)), x, 2)), print1(n^2, ", ")););} \\ Michel Marcus, May 05 2016

cp's OEIS Frontend

A163978 a(n) = 2*a(n-2) for n > 2; a(1) = 3, a(2) = 4.

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A166278 Square array A(n,k), n,k>=0, read by antidiagonals: A(n,k) is the total element sum of the k-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.

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A185333 Number of binary necklaces of n beads for which a cut exists producing a palindrome.

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A339893 a(n) = A000523(n) - A001222(n); floor(log_2(n)) minus the number of prime divisors of n, counted with multiplicity.

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A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).

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A364956 Numbers k such that A163511(k) is either k itself or its descendant in Doudna-tree, A005940 (or equally, in A163511-tree).

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A377000 Array read by ascending antidiagonals: T(n,k) = number of n-esthetic numbers with k digits.

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A258209 Numbers k for which A256999(A059893(k)) = k.

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