A003985 -id:A003985 - cp's OEIS frontend

A059895 Table a(i,j) = product prime[k]^(Ei[k] AND Ej[k]) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; AND is the bitwise operation on binary representation of the exponents.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1

Offset: 1

Marc LeBrun, Feb 06 2001

Analogous to GCD, with AND replacing MIN.

			The top left 18 X 18 corner of the array:
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1,  2,  1,  1,  1,  2,  1,  2,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2
1,  1,  3,  1,  1,  3,  1,  1,  1,  1,  1,  3,  1,  1,  3,  1,  1,  1
1,  1,  1,  4,  1,  1,  1,  4,  1,  1,  1,  4,  1,  1,  1,  1,  1,  1
1,  1,  1,  1,  5,  1,  1,  1,  1,  5,  1,  1,  1,  1,  5,  1,  1,  1
1,  2,  3,  1,  1,  6,  1,  2,  1,  2,  1,  3,  1,  2,  3,  1,  1,  2
1,  1,  1,  1,  1,  1,  7,  1,  1,  1,  1,  1,  1,  7,  1,  1,  1,  1
1,  2,  1,  4,  1,  2,  1,  8,  1,  2,  1,  4,  1,  2,  1,  1,  1,  2
1,  1,  1,  1,  1,  1,  1,  1,  9,  1,  1,  1,  1,  1,  1,  1,  1,  9
1,  2,  1,  1,  5,  2,  1,  2,  1, 10,  1,  1,  1,  2,  5,  1,  1,  2
1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 11,  1,  1,  1,  1,  1,  1,  1
1,  1,  3,  4,  1,  3,  1,  4,  1,  1,  1, 12,  1,  1,  3,  1,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 13,  1,  1,  1,  1,  1
1,  2,  1,  1,  1,  2,  7,  2,  1,  2,  1,  1,  1, 14,  1,  1,  1,  2
1,  1,  3,  1,  5,  3,  1,  1,  1,  5,  1,  3,  1,  1, 15,  1,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 16,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 17,  1
1,  2,  1,  1,  1,  2,  1,  2,  9,  2,  1,  1,  1,  2,  1,  1,  1, 18
A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 AND 3)* 3^(3 AND 5) = 2^1*3^1 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array

Cf. A003985 (A004198), A003989, A028234, A059896, A059897, A067029, A267115, A284578.

a[i_, i_] := i;
a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, Scan[(e1[#[[1]]] = #[[2]])&, f1]; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitAnd[e1[#], e2[#]]& /@ Intersection[f1[[All, 1]], f2[[All, 1]]]) ];
Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

(define (A059895 n) (A059895bi (A002260 n) (A004736 n)))
(define (A059895bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) m) ((= 1 b) m) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A004198bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (A028234 a) b m)) (else (loop a (A028234 b) m)))))
;; Antti Karttunen, Apr 11 2017

From Antti Karttunen, Apr 11 2017: (Start)
A(x,y) = A059896(x,y) / A059897(x,y).
A(x,y) * A059896(x,y) = A(x,y)^2 * A059897(x,y) = x*y.
(End)

Data section extended to 120 terms by Antti Karttunen, Apr 11 2017

A006581 a(n) = Sum_{k=1..n-1} (k AND n-k).

Original entry on oeis.org

1, 0, 4, 4, 5, 0, 12, 16, 21, 16, 24, 20, 17, 0, 32, 48, 65, 64, 84, 84, 85, 64, 92, 96, 101, 80, 88, 68, 49, 0, 80, 128, 177, 192, 244, 260, 277, 256, 316, 336, 357, 336, 360, 340, 321, 256, 336, 368, 401, 384, 420, 404, 389, 320, 364, 352, 341, 272, 264, 196

Offset: 2

N. J. A. Sloane

nonn

Marc LeBrun, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 2..8191
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 6, 24, 38-39, 64.
M. Le Brun, Email to N. J. A. Sloane, Jul 1991

Cf. A090889(n) - A000292(n-2).

Antidiagonal sums of array A003985.

Array[Sum[BitAnd[k, # - k], {k, # - 1}] &, 60, 2] (* Michael De Vlieger, Oct 27 2022 *)

def A006581(n): return (sum(k&n-k for k in range(1,n+1>>1))<<1)+(0 if n&1 else n>>1) # Chai Wah Wu, May 07 2023

G.f.: 1/(1-x)^2 * Sum_{k>=0} 2^k*t^2/(1+t)^2, t = x^2^k. - Ralf Stephan, Feb 12 2003
a(0) = a(1) = 0, a(2n) = 2*a(n-1) + 2*a(n) + n, a(2n+1) = 4*a(n).
a(n) = 2*(Sum_{k=1..floor((n-1)/2)} k AND n-k) + m where m = 0 if n is odd and n/2 otherwise. - Chai Wah Wu, May 07 2023

A099026 Array x AND NOT y, read by rising antidiagonals.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 2, 1, 0, 4, 2, 0, 0, 0, 5, 4, 1, 0, 1, 0, 6, 4, 4, 0, 2, 0, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 6, 4, 4, 0, 2, 0, 0, 0, 9, 8, 5, 4, 1, 0, 1, 0, 1, 0, 10, 8, 8, 4, 2, 0, 0, 0, 2, 0, 0, 11, 10, 9, 8, 3, 2, 1, 0, 3, 2, 1, 0, 12, 10, 8, 8, 8, 2, 0, 0, 4, 2, 0, 0, 0, 13, 12, 9, 8, 9, 8, 1

Offset: 0

Ralf Stephan, Sep 26 2004

For n>0, the n-th row and the differences of the n-th column have period 2^floor(log_(n)+1).

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

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10 (the color at (x, y) is function of T(x, y))

Rows include A000004, A059841. Columns include A001477, A052928. Antidiagonal sums are in A099027.

Cf. A003985 (AND), A003986 (OR), A003987 (XOR).

Table[BitAnd[x - y, BitNot[y]], {x, 0, 15}, {y, 0, x}] (* Paolo Xausa, Sep 30 2024 *)

PARI
```
T(x,y)=bitnegimply(x,y)
```

T(x, y) = x AND NOT y. The AND NOT operation satisfies the bitwise truth table: (0, 0) = 0, (0, 1) = 0, (1, 0) = 1, (1, 1) = 0.

cp's OEIS Frontend

A059895 Table a(i,j) = product prime[k]^(Ei[k] AND Ej[k]) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; AND is the bitwise operation on binary representation of the exponents.

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A006581 a(n) = Sum_{k=1..n-1} (k AND n-k).

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A099026 Array x AND NOT y, read by rising antidiagonals.

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Author

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Crossrefs

Programs

Mathematica

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