A004198 -id:A004198 - cp's OEIS frontend

A324658 a(n) = n - A324659(n), where A324659(n) is half of bitwise-AND of 2*n and sigma(n).

1, 2, 1, 4, 4, 0, 3, 8, 9, 2, 9, 0, 8, 2, 3, 16, 16, 0, 17, 0, 5, 4, 19, 0, 16, 10, 11, 0, 16, 26, 15, 32, 33, 32, 35, 0, 36, 32, 35, 0, 40, 10, 41, 4, 8, 10, 39, 0, 33, 16, 19, 4, 36, 2, 19, 0, 17, 18, 33, 40, 32, 14, 11, 64, 65, 2, 65, 64, 69, 6, 67, 8, 72, 66, 65, 8, 77, 10, 71, 0, 65, 64, 81, 4, 65, 20, 67, 0, 80

Offset: 1

Antti Karttunen, Mar 14 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A000203, A004198, A318468, A324648, A324659.

Cf. A324652 (positions of zeros).

Array[# - BitAnd[2*#, DivisorSigma[1, #]]/2 &, 100] (* Paolo Xausa, Mar 13 2024 *)

A324658(n) = (n-(bitand(2*n,sigma(n))/2));

a(n) = n - A324659(n) = n - A318468(n)/2 = n - ((2*n AND sigma(n))/2).

A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

Original entry on oeis.org

0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 3, 8, 9, 2, 3, 0, 1, 2, 1, 0, 1, 2, 3, 0, 0, 10, 1, 8, 5, 2, 1, 0, 1, 2, 3, 32, 1, 6, 1, 0, 9, 2, 1, 8, 33, 2, 3, 0, 0, 2, 3, 0, 1, 6, 3, 0, 1, 2, 3, 8, 1, 2, 33, 0, 1, 2, 65, 0, 1, 2, 3, 0, 1, 2, 1, 12, 1, 10, 5, 0, 16, 2, 3, 0, 1, 18, 7, 0, 1, 2, 9, 8, 1, 2, 7, 0, 1, 2, 35, 0, 65, 2, 33

Offset: 1

Antti Karttunen, Feb 19 2022

Cf. A000203, A004198, A007947, A019565, A080398, A351557, A351560.

Cf. also A324644, A351558.

Table[If[# == 1, 0, Total[#2*2^PrimePi[#1] & @@@ FactorInteger[#]]/2] &@ GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 103}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351559(n) = A048675(gcd(sigma(n), A019565(n)));

a(n) = A048675(A351557(n)) = A048675(gcd(sigma(n), A019565(n))).

a(n) = n AND A351560(n), where AND is bitwise-and, A004198.

A082856 Recursive binary interleaving code for rooted plane binary trees, as ordered by A014486.

Original entry on oeis.org

0, 1, 3, 5, 11, 35, 7, 21, 69, 139, 2059, 43, 547, 8227, 15, 39, 23, 277, 4117, 71, 85, 1093, 16453, 32907, 8388747, 2187, 526347, 134219787, 171, 2091, 555, 131619, 33554979, 8235, 8739, 2105379, 536879139, 143, 2063, 47, 551, 8231, 31, 55, 279, 65813, 16777493, 4119, 4373, 1052693, 268439573, 79, 103, 87, 341, 4181, 1095, 1109

Offset: 0

Antti Karttunen, May 06 2003

nonn

This encoding has a property that the greatest common subtree i.e. the intersect (or the least common supertree, the union) of any two trees can be obtained by simply computing the binary-AND (A004198) (or respectively: binary-OR, A003986) of the corresponding codes. See A082858-A082860.

			The empty tree . has code 0, the tree of two edges (and leaves) \/ has code 1 and in general tree's code is obtained by interleaving into odd and even bits (above bit-0, which is always 1 for nonempty trees) the codes for the left and right hand side subtrees of the tree.

Antti Karttunen, Alternative Catalan Orderings (with the complete Scheme source)

Inverse: A082857. Cf. A072634-A072637, A075173-A075174, A000695.

A285115 Row sums of A285118: a(n) = Sum_{k=1..(n-1)} (C(n-1,k-1) bitwise-and C(n-1,k)), a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 1, 0, 5, 8, 12, 20, 49, 64, 304, 672, 1204, 2648, 3852, 9320, 18297, 32960, 75472, 146392, 304920, 577336, 1211144, 2034072, 4801892, 7637392, 18795944, 33811680, 71566612, 139144320, 285508328, 569229920, 1069209737, 2314296064, 4167725024, 8567738280, 16894013736, 33135107200, 68279466472, 121133055024

Offset: 0

Antti Karttunen, Apr 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..256

Cf. A000079, A004198, A007318, A285113, A285114, A285118.

a[n_]:=If[n<2, 0, Sum[BitAnd[Binomial[n - 1,k - 1], Binomial[n - 1, k]], {k, n - 1}]]; Table[a[n], {n, 0, 100}] (* Indranil Ghosh, Apr 16 2017 *)

A285115(n) = if(n<2,0,sum(k=1,(n-1),bitand(binomial(n-1,k-1),binomial(n-1,k))));

(define (A285115 n) (add A285118 (A000217 n) (+ -1 (A000217 (+ 1 n)))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))

a(0) = a(1) = 0, and for n > 1, a(n) = Sum_{k=1..(n-1)} C(n-1,k-1) AND C(n-1,k), where C(n,k) is a binomial coefficient & AND is bitwise-AND (A004198).
a(n) = A285113(n) - A285114(n).
a(n) = A000079(n) - A285113(n) = (A000079(n) - A285114(n))/2.

A324648 a(n) = n - A318458(n), where A318458(n) is bitwise-AND of n and the sum of proper divisors of n (sigma(n)-n).

Original entry on oeis.org

1, 2, 2, 4, 4, 0, 6, 8, 9, 2, 10, 12, 12, 4, 6, 16, 16, 2, 18, 0, 20, 16, 22, 24, 25, 10, 18, 0, 28, 20, 30, 32, 32, 34, 34, 0, 36, 32, 38, 8, 40, 8, 42, 4, 12, 36, 46, 48, 49, 16, 34, 16, 52, 52, 38, 56, 40, 26, 58, 16, 60, 28, 22, 64, 64, 0, 66, 68, 68, 4, 70, 0, 72, 66, 74, 12, 76, 4, 78, 16, 81, 82, 82, 80, 64, 80, 86, 0, 88

Offset: 1

Antti Karttunen, Mar 14 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A001065, A004198, A318458, A324658, A324649 (positions of zeros).

Array[# - BitAnd[#, DivisorSigma[1, #] - #] &, 100] (* Paolo Xausa, Mar 12 2024 *)

A318458(n) = bitand(n,sigma(n)-n);
A324648(n) = (n-A318458(n));

PARI
```
A324648(n) = (n-bitand(n,sigma(n)-n));
```

a(n) = n - A318458(n).

A324659 a(n) = (1/2)A318468(n), where A318468(n) is bitwise-AND of 2n and sigma(n).

Original entry on oeis.org

0, 0, 2, 0, 1, 6, 4, 0, 0, 8, 2, 12, 5, 12, 12, 0, 1, 18, 2, 20, 16, 18, 4, 24, 9, 16, 16, 28, 13, 4, 16, 0, 0, 2, 0, 36, 1, 6, 4, 40, 1, 32, 2, 40, 37, 36, 8, 48, 16, 34, 32, 48, 17, 52, 36, 56, 40, 40, 26, 20, 29, 48, 52, 0, 0, 64, 2, 4, 0, 64, 4, 64, 1, 8, 10, 68, 0, 68, 8, 80, 16, 18, 2, 80, 20, 66, 20, 88, 9, 80

Offset: 1

Antti Karttunen, Mar 14 2019

Cf. A000203, A004198, A318458, A318468, A324648, A324658.

Cf. A324652 (fixed points).

Array[BitAnd[2*#, DivisorSigma[1, #]]/2 &, 100] (* Paolo Xausa, Mar 11 2024 *)

PARI
```
A324659(n) = (bitand(2*n,sigma(n))/2);
```

a(n) = A318468(n)/2.

a(n) = n - A324658(n).

A334348 The terms in the Zeckendorf representation of T(n, k) correspond to the terms in common in the Zeckendorf representations of n and of k; square array T(n, k) read by antidiagonals, n, k >= 0.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Apr 24 2020

This array has connections with the bitwise AND operator (A004198).

			Square array 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  0  1  0  1  0  0  1   0   0   1   0
    2|  0  0  2  0  0  0  0  2  0  0   2   0   0   0
    3|  0  0  0  3  3  0  0  0  0  0   0   3   3   0
    4|  0  1  0  3  4  0  1  0  0  1   0   3   4   0
    5|  0  0  0  0  0  5  5  5  0  0   0   0   0   0
    6|  0  1  0  0  1  5  6  5  0  1   0   0   1   0
    7|  0  0  2  0  0  5  5  7  0  0   2   0   0   0
    8|  0  0  0  0  0  0  0  0  8  8   8   8   8   0
    9|  0  1  0  0  1  0  1  0  8  9   8   8   9   0
   10|  0  0  2  0  0  0  0  2  8  8  10   8   8   0
   11|  0  0  0  3  3  0  0  0  8  8   8  11  11   0
   12|  0  1  0  3  4  0  1  0  8  9   8  11  12   0
   13|  0  0  0  0  0  0  0  0  0  0   0   0   0  13

Rémy Sigrist, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150)
Rémy Sigrist, Colored representation of (x, y) for 0 <= x, y <= 1000 (where the hue is function of T(x, y))
Rémy Sigrist, PARI program for A334348
Wikipedia, Zeckendorf's theorem
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003714, A022290, A004198, A332022, A332565.

PARI
```
See Links section.
```

T(n, k) = A022290(A003714(n) AND A003714(k)) (where AND denotes the bitwise AND operator, A004198).
T(n, 0) = 0.
T(n, n) = n.
T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).

A351558 a(n) = A048675(gcd(n, A019565(n))).

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 8, 6, 0, 0, 2, 0, 4, 0, 16, 0, 0, 0, 0, 2, 8, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 34, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 0, 2, 32, 0, 2, 20, 8, 0, 0, 0, 4, 0, 0, 10, 0, 0, 2, 0, 64, 0, 4, 0, 0, 0, 0, 2, 0, 8, 2, 0, 0, 0, 0, 0, 0, 68, 0, 2, 16, 0, 2, 8, 0, 0, 0, 4, 0, 0, 0, 2, 4, 0, 66

Offset: 0

Antti Karttunen, Feb 19 2022

Cf. A004198, A007947, A019565, A087207, A351556.

Cf. also A324198, A351559.

Table[If[# == 1, 0, Total[#2*2^PrimePi[#1] & @@@ FactorInteger[#]]/2] &@ GCD[n, Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 102}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351558(n) = A048675(gcd(n, A019565(n)));

a(n) = A048675(A351556(n)) = A048675(gcd(n, A019565(n))).

a(n) = n AND A087207(n), where AND is bitwise-and, A004198.

A353296 Pairs (i,j) of positive integers with at least one common 1-bit sorted first by i+j then by i.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 3, 1, 2, 3, 3, 2, 1, 5, 3, 3, 5, 1, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 1, 9, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 9, 1, 4, 7, 5, 6, 6, 5, 7, 4, 1, 11, 2, 10, 3, 9, 5, 7, 6, 6, 7, 5, 9, 3, 10, 2, 11, 1

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Apr 09 2022

Pairs (i,j) with AND(i,j) <> 0.
See A352909 for the other pairs.
There are A048967(n) pairs (i,j) with n = i+j.

			The first pairs are:
    [1, 1],
    [1, 3], [2, 2], [3, 1],
    [2, 3], [3, 2],
    [1, 5], [3, 3], [5, 1],
    [1, 7], [2, 6], [3, 5], [4, 4], [5, 3], [6, 2], [7, 1],
    [2, 7], [3, 6], [4, 5], [5, 4], [6, 3], [7, 2],
    [1, 9], [3, 7], [4, 6], [5, 5], [6, 4], [7, 3], [9, 1],
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..12392 (values for i+j <= 128)

Cf. A004198, A048967, A352909, A353297 (i values), A353298 (j values).

A353296row[ij_] := Select[Array[{#, ij-#} &, ij], BitAnd @@ # > 0 &];
Array[A353296row, 15] (* Paolo Xausa, Feb 24 2024 *)

for (ij=1, 12, for (i=1, ij, j=ij-i; if (bitand(i,j), print1(i", "j", "))))

A213540 Numbers k such that k AND k*2 = 2, where AND is the bitwise AND operator.

Original entry on oeis.org

3, 11, 19, 35, 43, 67, 75, 83, 131, 139, 147, 163, 171, 259, 267, 275, 291, 299, 323, 331, 339, 515, 523, 531, 547, 555, 579, 587, 595, 643, 651, 659, 675, 683, 1027, 1035, 1043, 1059, 1067, 1091, 1099, 1107, 1155, 1163, 1171, 1187, 1195, 1283, 1291, 1299, 1315

Offset: 1

Alex Ratushnyak, Jun 14 2012

			In binary, 19 is 10011, while 2 * 19 = 38 is of course 100110. Since 010011 AND 100110 = 000010 (in decimal, 2), 19 is in the sequence.
20 is not in the sequence, since 010100 AND 101000 = 000000.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A213370, A004198, A003714.

F:= combinat[fibonacci]:
b:= proc(n) local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= n-> 8*b(n-1)+3:
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 17 2012

Select[Range[1024], BitAnd[#, 2#] == 2 &] (* Alonso del Arte, Jun 18 2012 *)

is(n)=bitand(n,2*n)==2 \\ Charles R Greathouse IV, Jun 18 2012

for n in range(1777):
    a = 2*n & n
    if a==2:
        print(n, end=',')

a(n) = A003714(n-1)*8 + 3.

cp's OEIS Frontend

A324658 a(n) = n - A324659(n), where A324659(n) is half of bitwise-AND of 2*n and sigma(n).

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A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

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A082856 Recursive binary interleaving code for rooted plane binary trees, as ordered by A014486.

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A285115 Row sums of A285118: a(n) = Sum_{k=1..(n-1)} (C(n-1,k-1) bitwise-and C(n-1,k)), a(0) = a(1) = 0.

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A324648 a(n) = n - A318458(n), where A318458(n) is bitwise-AND of n and the sum of proper divisors of n (sigma(n)-n).

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PARI

Formula

A324659 a(n) = (1/2)*A318468(n), where A318468(n) is bitwise-AND of 2*n and sigma(n).

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A334348 The terms in the Zeckendorf representation of T(n, k) correspond to the terms in common in the Zeckendorf representations of n and of k; square array T(n, k) read by antidiagonals, n, k >= 0.

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Programs

PARI

Formula

A351558 a(n) = A048675(gcd(n, A019565(n))).

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Author

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Programs

Mathematica

A324659 a(n) = (1/2)A318468(n), where A318468(n) is bitwise-AND of 2n and sigma(n).