A004198 -id:A004198 - cp's OEIS frontend

A221146 Table read by antidiagonals: (m+n) - (m XOR n).

0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 4, 4, 0, 0, 0, 2, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 8, 2, 4, 2, 0, 0, 0, 4, 4, 8, 8, 4, 4, 0, 0, 0, 2, 0, 6, 8, 10, 8, 6, 0, 2, 0, 0, 0, 0, 0, 8, 8, 8, 8, 0, 0, 0, 0, 0, 2, 4, 2, 0, 10, 12, 10, 0, 2, 4, 2, 0

Offset: 0

BOCUT Adrian Sebastian, Dec 12 2012

Equals twice A004198.
This sequence is related to two fractals: the Sierpinski gasket fractal and Peano filigree.
For the Sierpinski fractal the procedure is the following:
- write the number stored in the position (i,j) as i+j + d, where d stands for difference.
The array of the differences is
  0 0 0 0
  0 2 0 2
  0 0 4 4
  0 2 4 6
If this matrix is represented by colors we obtain the Sierpinski gasket; coordinates (i,j) contain a pixel with the color i XOR j.
If we follow the odd and even numbers of the XOR table we obtain the Peano curve.

			Table begins:
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0 ...
0   2   0   2   0   2   0   2   0   2   0   2   0   2   0   2 ...
0   0   4   4   0   0   4   4   0   0   4   4   0   0   4   4 ...
0   2   4   6   0   2   4   6   0   2   4   6   0   2   4   6 ...
0   0   0   0   8   8   8   8   0   0   0   0   8   8   8   8 ...
0   2   0   2   8  10   8  10   0   2   0   2   8  10   8  10 ...
0   0   4   4   8   8  12  12   0   0   4   4   8   8  12  12 ...
0   2   4   6   8  10  12  14   0   2   4   6   8  10  12  14 ...
0   0   0   0   0   0   0   0  16  16  16  16  16  16  16  16 ...
0   2   0   2   0   2   0   2  16  18  16  18  16  18  16  18 ...
0   0   4   4   0   0   4   4  16  16  20  20  16  16  20  20 ...
0   2   4   6   0   2   4   6  16  18  20  22  16  18  20  22 ...
0   0   0   0   8   8   8   8  16  16  16  16  24  24  24  24 ...
0   2   0   2   8  10   8  10  16  18  16  18  24  26  24  26 ...
0   0   4   4   8   8  12  12  16  16  20  20  24  24  28  28 ...
0   2   4   6   8  10  12  14  16  18  20  22  24  26  28  30 ...
...

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Cf. A003987, A004198.

Table[m-BitXor[n, m-n], {m, 0, 15}, {n, 0, m}] (* Paolo Xausa, Mar 14 2024 *)

Edited by N. J. A. Sloane, Jan 03 2013

A256754 a(n) = bitwise AND of n and the reverse of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 4, 13, 8, 3, 16, 1, 16, 19, 0, 4, 22, 0, 8, 16, 26, 8, 16, 28, 2, 13, 0, 33, 34, 33, 36, 1, 2, 5, 0, 8, 8, 34, 44, 36, 0, 10, 16, 16, 0, 3, 16, 33, 36, 55, 0, 9, 16, 27, 4, 16, 26, 36, 0, 0, 66, 64, 68, 64, 6, 1, 8, 1, 10

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A004086, A004198, A055483, A056964, A056965, A061205, A068634, A256755, A256756.

a:= n-> Bits[And](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitAnd[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,74}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitand(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A004198(n,A004086(n)).

A336882 a(0) = 1; for k >= 0, 0 <= i < 2^k, a(2^k + i) = m_k * a(i), where m_k is the least odd number not in terms 0..2^k - 1.

Original entry on oeis.org

1, 3, 5, 15, 7, 21, 35, 105, 9, 27, 45, 135, 63, 189, 315, 945, 11, 33, 55, 165, 77, 231, 385, 1155, 99, 297, 495, 1485, 693, 2079, 3465, 10395, 13, 39, 65, 195, 91, 273, 455, 1365, 117, 351, 585, 1755, 819, 2457, 4095, 12285, 143, 429, 715, 2145, 1001

Offset: 0

Peter Munn, Aug 16 2020

nonn

A permutation of the odd numbers.
Every positive integer, m, is the product of a unique subset of the terms of A050376. The members of the subset are often known as the Fermi-Dirac factors of m. In this sequence, the odd numbers appear lexicographically according to their Fermi-Dirac factors (with those factors listed in decreasing order). The equivalent sequence for all positive integers is A052330.
The sequence has a conditional exponential identity shown in the formula section. This relies on the offset being 0, as in related sequences, notably A019565 and A052330.

			a(0) = 1, as specified explicitly.
m_0 = 3, the least odd number not in terms 0..0.
So a(1) = a(2^0 + 0) = m_0 * a(0) = 3 * 1 = 3.
m_1 = 5, the least odd number not in terms 0..1.
So a(2) = a(2^1 + 0) = m_1 * a(0) = 5 * 1 = 5;
and a(3) = a(2^1 + 1) = m_1 * a(1) = 5 * 3 = 15.
The initial terms are tabulated below, equated with the product of their Fermi-Dirac factors to exhibit the lexicographic order. We start with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
   n     a(n)
   0    1,
   1    3 = 3,
   2    5 = 5,
   3   15 = 5 * 3,
   4    7 = 7,
   5   21 = 7 * 3,
   6   35 = 7 * 5,
   7  105 = 7 * 5 * 3,
   8    9 = 9,
   9   27 = 9 * 3,
  10   45 = 9 * 5,
  11  135 = 9 * 5 * 3,
  12   63 = 9 * 7.

Sean A. Irvine, Java program (github)

Permutation of A005408.
Subsequence of A052330.
Subsequences: A062090, A332382 (squarefree terms).
A003986, A003987, A004198, A059896, A059897 are used to express relationship between terms of this sequence.
Cf. A019565, A050376.

a(2^k) = min({ 2*m+1 : m >= 0, 2*m+1 <> a(j), 0 <= j < 2^k }) = A062090(k+2).
If x AND y = 0, a(x+y) = a(x) * a(y), where AND denotes the bitwise operation, A004198(.,.).
a(x XOR y) = A059897(a(x), a(y)), where XOR denotes bitwise exclusive-or, A003987(.,.).
a(x OR y) = A059896(a(x), a(y)), where OR denotes the bitwise operation, A003986(.,.).

A356325 Array A(n, k), n, k >= 0, read by antidiagonals; the terms in the negaFibonacci representation of A(n, k) are the terms in common in the negaFibonacci representations of n and k.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 2, 1, 5, 5, 1, 2, 1, 0, 0, 0, 2, 2, 5, 5, 5, 2, 2, 0, 0, 0, 0, 0, 3, 5, 5, 5, 5, 3, 0, 0, 0, 0, 1, 0, 0, 5, 5, 6, 5, 5, 0, 0, 1, 0

Offset: 0

Rémy Sigrist, Aug 03 2022

This sequence has similarities with A334348.

			Array A(n, k) begins:
  n\k|  0  1  2  3   4  5  6  7  8   9  10  11  12  13
  ---+------------------------------------------------
    0|  0  0  0  0   0  0  0  0  0   0   0   0   0   0
    1|  0  1  0  1   0  0  1  0  1   0   0   1   0   0
    2|  0  0  2  2   0  0  0  2  2   0   0   0   0   0
    3|  0  1  2  3   0  0  1  2  3   0   0   1   0   0
    4|  0  0  0  0   4  5  5  5  5  -1   0   0  -1   0
    5|  0  0  0  0   5  5  5  5  5   0   0   0   0   0
    6|  0  1  0  1   5  5  6  5  6   0   0   1   0   0
    7|  0  0  2  2   5  5  5  7  7   0   0   0   0   0
    8|  0  1  2  3   5  5  6  7  8   0   0   1   0   0
    9|  0  0  0  0  -1  0  0  0  0   9  10  10  12  13
   10|  0  0  0  0   0  0  0  0  0  10  10  10  13  13
   11|  0  1  0  1   0  0  1  0  1  10  10  11  13  13
   12|  0  0  0  0  -1  0  0  0  0  12  13  13  12  13
   13|  0  0  0  0   0  0  0  0  0  13  13  13  13  13
.
For n = 14 and k = 43:
- using F(-k) = A039834(k):
- 14 = F(-1) + F(-7),
- 43 = F(-2) + F(-4) + F(-7) + F(-9),
- so A(14, 43) = F(-7) = 13.

Rémy Sigrist, Colored representation of the array for n, k <= 1000 (white for 0's, shades of blue for negative values, shades of red for positive values)
Rémy Sigrist, PARI program
Wikipedia, NegaFibonacci coding
Index entries for sequences related to Zeckendorf expansion of n

Cf. A004198, A039834, A215024, A309076, A334348, A356326, A356327.

PARI
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See Links section.
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A(n, k) = A(k, n).
A(n, n) = n.
A(n, 0) = 0.
A(n, k) = A356327(A215024(n) AND A215024(k)) (where AND denotes the bitwise AND operator).

A261078 Semiprimes p*q such that q = p + 2^k for some k >= 0.

Original entry on oeis.org

6, 15, 21, 33, 35, 57, 65, 77, 143, 161, 185, 201, 209, 221, 323, 377, 393, 437, 473, 497, 713, 899, 1073, 1457, 1517, 1529, 1577, 1763, 1769, 1841, 1961, 2021, 2537, 2993, 3233, 3473, 3497, 3599, 3713, 3737, 3953, 4553, 4601, 4757, 5183, 5561, 5609, 5753, 6497, 6557, 7217, 7313, 8633, 8777, 9593, 9797, 10001, 10265, 10403, 10841, 10961, 11009, 11021

Offset: 1

Antti Karttunen, Sep 22 2015

nonn

Terms ending with digit 5 (in decimal) are very rare, because terms of A123250 are rare.

			6 = 2*3 is present as 3 = 2 + 2^0.
15 = 3*5 is present as 5 = 3 + 2^1.
35 = 5*7 is present as 7 = 5 + 2^1.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001222, A004198, A006530, A020639, A123250.

Cf. also A261073, A261077 (subsequences).

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
isA261078(n) = { my(d); if(bigomega(n)!=2, return(0), d = (n/A020639(n)) - A020639(n); (d && !bitand(d,d-1))); };
i=0; n=0; while(i < 10000, n++; if(isA261078(n), i++; write("b261078.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A261078 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (pow2? (- (A006530 n) (A020639 n)))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1))))) ;; A004198bi implements bitwise-AND (Cf. A004198)

A268040 Array y AND NOT x, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 2, 3, 0, 0, 0, 2, 4, 0, 1, 0, 1, 4, 5, 0, 0, 2, 0, 4, 4, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 2, 0, 4, 4, 6, 8, 0, 1, 0, 1, 0, 1, 4, 5, 8, 9, 0, 0, 2, 0, 0, 0, 2, 4, 8, 8, 10, 0, 1, 2, 3, 0, 1, 2, 3, 8, 9, 10, 11, 0, 0, 0, 2, 4, 0, 0, 2, 8, 8, 8, 10, 12

Offset: 0

Peter Kagey, Jan 24 2016

This is one of eight bitwise truth tables subject to the condition that (0, 0) implies 0.
#        | A000004 | A004198 | A099026 | A025581
# -------|---------|---------|---------|---------
# (0, 0) | 0       | 0       | 0       | 0
# (0, 1) | 0       | 0       | 0       | 0
# (1, 0) | 0       | 0       | 1       | 1
# (1, 1) | 0       | 1       | 0       | 1
#
#        | A268040 | A002262 | A003987 | A003986
# -------|---------|---------|---------|---------
# (0, 0) | 0       | 0       | 0       | 0
# (0, 1) | 1       | 1       | 1       | 1
# (1, 0) | 0       | 0       | 1       | 1
# (1, 1) | 0       | 1       | 0       | 1
Transpose of the array in A099026. - Michel Marcus, Jan 25 2016

			The array begins:
0, 1, 2, 3, 4, 5, ...
0, 0, 2, 2, 4, 4, ...
0, 1, 0, 1, 4, 5, ...
0, 0, 0, 0, 4, 4, ...
0, 1, 2, 3, 0, 1, ...
0, 0, 2, 2, 0, 0, ...
...

Peter Kagey, Table of n, a(n) for n = 0..8191

Cf. A000004, A004198, A099026, A025581, A003987, A002262, A003986.

T(x, y)=bitnegimply(y, x) \\ Michel Marcus, Jan 25 2016

A285720 Number of ways to write n as a sum of two unordered squarefree numbers so that their addition in base-2 does not produce carries.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 11, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 11, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 7, 0, 0, 0, 13, 0, 0, 0, 3, 0, 0, 0, 9, 0

Offset: 1

Antti Karttunen, May 02 2017

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences related to binary expansion of n

Cf. A003987, A004198, A005117, A008683, A008966, A013928, A071068, A088512.

Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]] Boole[BitXor[i, n - i] == n], {i, Floor[n/2]}], {n, 120}] (* Michael De Vlieger, May 03 2017 *)

from sympy import mobius
def a003987(n, i): return i^(n - i) == n
def a(n): return sum([abs(mobius(i)*mobius(n - i))*(1*a003987(n, i)) for i in range(1, n//2 + 1)])
print([a(n) for n in range(1,121)]) # Indranil Ghosh, May 02 2017

(define (A285720 n) (let loop ((k (A013928 n)) (s 0)) (if (or (zero? k) (< (A005117 k) (- n (A005117 k)))) s (loop (- k 1) (+ s (if (and (= 1 (A008966 (- n (A005117 k)))) (zero? (A004198bi (A005117 k) (- n (A005117 k))))) 1 0)))))) ;; Where A004198bi implements bitwise-AND (A004198).

a(n) = Sum_{i=1..floor(n/2)} abs(mu(i)*mu(n-i))*[A003987(i,n-i) == n]. (Here [] is Iverson bracket, giving in this case 1 only if (i XOR (n-i)) is equal to n, and 0 otherwise. mu is Moebius mu function, A008683.)
a(n) <= A071068(n).
a(n) <= A088512(n).

A335587 a(n) is the sum of the numbers k such that 0 <= k <= n and n AND k = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 0, 6, 2, 1, 0, 28, 12, 10, 4, 6, 2, 1, 0, 120, 56, 52, 24, 44, 20, 18, 8, 28, 12, 10, 4, 6, 2, 1, 0, 496, 240, 232, 112, 216, 104, 100, 48, 184, 88, 84, 40, 76, 36, 34, 16, 120, 56, 52, 24, 44, 20, 18, 8, 28, 12, 10, 4, 6, 2, 1, 0, 2016, 992, 976, 480

Offset: 0

Rémy Sigrist, Apr 21 2021

All terms can be written as m * 2^A000120(m) for some m >= 0.

			For n = 4:
- 4 AND 0 = 0,
- 4 AND 1 = 0,
- 4 AND 2 = 0,
- 4 AND 3 = 0,
- 4 AND 4 = 4,
- so a(4) = 0 + 1 + 2 + 3 = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Colored logarithmic scatterplot of the sequence for n < 2^16 (where the color is function of A080791(n))
Index entries for sequences related to binary expansion of n

Cf. A000120, A004198 (bitwise AND), A006516, A035327, A080100, A080791.

a(n) = sum(k=0, n, if (bitand(n, k)==0, k, 0))

a(n) = my (w=#binary(n)); ( (2^w-1-n) * 2^(w-hammingweight(n)) ) \ 2

a(n) = A035327(n) * A080100(n) / 2 for any n > 0.
a(2*n+1) = 2*a(n).
a(2^k-1) = 0 for any k >= 0.
a(2^k) = A006516(k) for any k >= 0.

A339601 Starting from x_0 = n, iterate by dividing with 3 (discarding any remainder), until zero is reached: x_1 = floor(x_0/3), x_2 = floor(x_1/3), etc. Then a(n) = Sum_{i=0..} (x_i AND 2^i), where AND is bitwise-and.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, 3, 0, 1, 0, 1, 0, 1, 2, 3, 2, 3

Offset: 0

Antti Karttunen, Dec 09 2020

Index entries for sequences related to binary expansion of n

Cf. A004198, A037095.

Cf. also A332497.

Array[Total@ MapIndexed[BitAnd[2^First[#2 - 1], #1] &, NestWhileList[Floor[#/3] &, #, # > 0 &]] &, 106, 0] (* Michael De Vlieger, Dec 10 2020 *)

A339601(n) = { my(i=0, s=0); while(n, s += bitand(2^i,n); i++; n \= 3); (s); };

A339601(n) = { my(m=1, s=0); while(n>=m, s += bitand(m,n); m <<= 1; n \= 3); (s); };

A352727 Square array A(n, k), n, k >= 0, read by antidiagonals: the binary expansion of A(n, k) contains the runs of consecutive 1's that appear both in the binary expansions of n and k.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0

Offset: 0

Rémy Sigrist, Mar 30 2022

We only consider maximal runs of one or more consecutive 1's (as counted by A069010) that completely match in binary expansions of n and k, not simply single common 1's.

			Table A(n, k) begins:
  n\k|  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
  ---+------------------------------------------------------
    0|  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0
    1|  0  1  0  0  0  1  0  0  0  1   0   0   0   1   0   0
    2|  0  0  2  0  0  0  0  0  0  0   2   0   0   0   0   0
    3|  0  0  0  3  0  0  0  0  0  0   0   3   0   0   0   0
    4|  0  0  0  0  4  4  0  0  0  0   0   0   0   0   0   0
    5|  0  1  0  0  4  5  0  0  0  1   0   0   0   1   0   0
    6|  0  0  0  0  0  0  6  0  0  0   0   0   0   0   0   0
    7|  0  0  0  0  0  0  0  7  0  0   0   0   0   0   0   0
    8|  0  0  0  0  0  0  0  0  8  8   8   8   0   0   0   0
    9|  0  1  0  0  0  1  0  0  8  9   8   8   0   1   0   0
   10|  0  0  2  0  0  0  0  0  8  8  10   8   0   0   0   0
   11|  0  0  0  3  0  0  0  0  8  8   8  11   0   0   0   0
   12|  0  0  0  0  0  0  0  0  0  0   0   0  12  12   0   0
   13|  0  1  0  0  0  1  0  0  0  1   0   0  12  13   0   0
   14|  0  0  0  0  0  0  0  0  0  0   0   0   0   0  14   0
   15|  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10 (where the hue is function of T(n, k); black pixels denote 0's)
Index entries for sequences related to binary expansion of n

Cf. A004198, A069010, A352724, A352729.

A352724(n) = { my (r=[], o=0); while (n, my (v=valuation(n+n%2, 2)); if (n%2, r=concat(r, (2^v-1)*2^o)); o+=v; n\=2^v); r }
A(n,k) = vecsum(setintersect(A352724(n), A352724(k)))

A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, n) = n.
A(n, 2*n) = 0.
A(n, k) <= A004198(n, k) (bitwise AND operator).
A(n, n+1) = A352729(n).

cp's OEIS Frontend

A221146 Table read by antidiagonals: (m+n) - (m XOR n).

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A256754 a(n) = bitwise AND of n and the reverse of n.

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A336882 a(0) = 1; for k >= 0, 0 <= i < 2^k, a(2^k + i) = m_k * a(i), where m_k is the least odd number not in terms 0..2^k - 1.

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A356325 Array A(n, k), n, k >= 0, read by antidiagonals; the terms in the negaFibonacci representation of A(n, k) are the terms in common in the negaFibonacci representations of n and k.

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A261078 Semiprimes p*q such that q = p + 2^k for some k >= 0.

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A268040 Array y AND NOT x, read by antidiagonals.

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A285720 Number of ways to write n as a sum of two unordered squarefree numbers so that their addition in base-2 does not produce carries.

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A335587 a(n) is the sum of the numbers k such that 0 <= k <= n and n AND k = 0 (where AND denotes the bitwise AND operator).

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