A004198 -id:A004198 - cp's OEIS frontend

A286099 Square array read by antidiagonals: A(n,k) = T(n OR k, n AND k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and OR is bitwise-or (A003986).

0, 2, 2, 5, 4, 5, 9, 9, 9, 9, 14, 13, 12, 13, 14, 20, 20, 18, 18, 20, 20, 27, 26, 27, 24, 27, 26, 27, 35, 35, 35, 35, 35, 35, 35, 35, 44, 43, 42, 43, 40, 43, 42, 43, 44, 54, 54, 52, 52, 50, 50, 52, 52, 54, 54, 65, 64, 65, 62, 61, 60, 61, 62, 65, 64, 65, 77, 77, 77, 77, 73, 73, 73, 73, 77, 77, 77, 77, 90, 89, 88, 89, 90, 85, 84, 85, 90, 89, 88, 89, 90

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
   2,   4,   9,  13,  20,  26,  35,  43,  54,  64,  77,  89, 104
   5,   9,  12,  18,  27,  35,  42,  52,  65,  77,  88, 102, 119
   9,  13,  18,  24,  35,  43,  52,  62,  77,  89, 102, 116, 135
  14,  20,  27,  35,  40,  50,  61,  73,  90, 104, 119, 135, 148
  20,  26,  35,  43,  50,  60,  73,  85, 104, 118, 135, 151, 166
  27,  35,  42,  52,  61,  73,  84,  98, 119, 135, 150, 168, 185
  35,  43,  52,  62,  73,  85,  98, 112, 135, 151, 168, 186, 205
  44,  54,  65,  77,  90, 104, 119, 135, 144, 162, 181, 201, 222
  54,  64,  77,  89, 104, 118, 135, 151, 162, 180, 201, 221, 244
  65,  77,  88, 102, 119, 135, 150, 168, 181, 201, 220, 242, 267
  77,  89, 102, 116, 135, 151, 168, 186, 201, 221, 242, 264, 291
  90, 104, 119, 135, 148, 166, 185, 205, 222, 244, 267, 291, 312

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000096 (row 0 & column 0), A162761 (seems to be row 1 & column 1), A046092 (main diagonal).
Cf. A003056, A003986, A004198.
Cf. also arrays A286098, A286101, A286102, A286109.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitOr[n, k],BitAnd[n,  k]]; Table[A[n - k, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n|k, n&k)
for n in range(0, 21): print([A(k, n - k) for k in range(0, n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286099 n) (A286099bi (A002262 n) (A025581 n)))
(define (A286099bi row col) (let ((a (A003986bi row col)) (b (A004198bi row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003986bi and A004198bi implement bitwise-OR (A003986) and bitwise-AND (A004198).

A(n,k) = T(A003986(n,k), A004198(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A318463 a(n) = Sum_{d|n, d < n/d} (d AND n/d), where AND is bitwise-and (A004198).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 2, 0, 1, 2, 1, 6, 4, 2, 1, 4, 1, 0, 2, 6, 1, 8, 1, 0, 4, 0, 6, 2, 1, 2, 2, 0, 1, 8, 1, 2, 5, 2, 1, 4, 1, 0, 2, 6, 1, 4, 2, 4, 4, 0, 1, 12, 1, 2, 3, 0, 6, 4, 1, 2, 4, 8, 1, 12, 1, 0, 7, 2, 4, 8, 1, 12, 4, 0, 1, 16, 2, 2, 2, 12, 1, 16, 6, 6, 4, 2, 2, 8, 1, 6, 11, 6, 1, 4, 1, 8, 16

Offset: 1

Antti Karttunen, Aug 28 2018

Cf. A000203, A004198, A318458, A318460, A318461, A318462.

A318463(n) = { my(ands=0); fordiv(n,d,if(d<(n/d), ands += bitand(d,n/d))); (ands); };

a(n) = A000203(n) - A318461(n) = (A000203(n)-A318462(n))/2.

A318518 a(n) = A032742(n) AND n-A032742(n), where AND is bitwise-and (A004198) and A032742 = the largest proper divisor of n.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 4, 2, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 6, 11, 0, 12, 4, 13, 0, 14, 0, 15, 0, 16, 2, 17, 4, 18, 0, 19, 8, 20, 0, 21, 0, 22, 14, 23, 0, 24, 2, 25, 0, 26, 0, 27, 8, 28, 2, 29, 0, 30, 0, 31, 0, 32, 4, 33, 0, 34, 6, 35, 0, 36, 0, 37, 16, 38, 2, 39, 0, 40, 18, 41, 0, 42, 0, 43, 24, 44, 0, 45, 12, 46, 30, 47, 0, 48, 0, 49, 0, 50, 0, 51, 0

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004198, A032742, A060681, A318515, A318516, A318517.

Cf. also A318508.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318518(n) = bitand(A032742(n),n-A032742(n));

a(n) = A004198(A032742(n), A060681(n)).

a(n) = n - A318516(n) = (n - A318517(n))/2.

A283978 a(2n) = 0, a(2n+1) = A002487(n) AND A002487(n+1), where AND is bitwise-and (A004198).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 4, 0, 3, 0, 0, 0, 0, 0, 5, 0, 2, 0, 2, 0, 5, 0, 0, 0, 0, 0, 3, 0, 4, 0, 4, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 0, 2, 0, 2, 0, 3, 0, 8, 0, 8, 0, 5, 0, 4, 0, 4, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 4, 0, 5, 0, 8, 0, 8, 0, 3, 0, 2, 0, 2, 0, 3, 0, 0, 0

Offset: 0

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Bisections: A000004, A283988.

Cf. A002487, A004198, A283976, A283977.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[If[EvenQ@ n, 0, BitAnd[a[#], a[# + 1]] &[(n - 1)/2]], {n, 0, 120}] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
a(n) = if(n<2, 0, if(n%2, bitand(A(n\2), A((n + 1)/2)), 0));
for(n=0, 120, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 23 2017

(define (A283978 n) (if (even? n) 0 (A004198bi (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))) ;; Where A004198bi implements bitwise-AND (A004198).

a(2n) = 0, a(2n+1) = A002487(n) AND A002487(n+1), where AND is bitwise-and (A004198).
a(n) = A283976(n) - A283977(n).
a(n) = A002487(n) - A283976(n) = (A002487(n) - A283977(n))/2.

A285118 Triangle read by rows: T(0,n) = T(n,n) = 0; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) AND C(n-1,k), where C(n,k) is binomial coefficient (A007318) & AND is bitwise-AND (A004198).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 4, 4, 0, 0, 0, 1, 0, 10, 0, 1, 0, 0, 0, 6, 4, 4, 6, 0, 0, 0, 1, 5, 1, 35, 1, 5, 1, 0, 0, 0, 8, 24, 0, 0, 24, 8, 0, 0, 0, 1, 0, 4, 84, 126, 84, 4, 0, 1, 0, 0, 0, 8, 40, 80, 208, 208, 80, 40, 8, 0, 0, 0, 1, 3, 37, 0, 330, 462, 330, 0, 37, 3, 1, 0, 0, 0, 0, 64, 204, 264, 792, 792, 264, 204, 64, 0, 0, 0

Offset: 0

Antti Karttunen, Apr 16 2017

			Rows 0-13 of array:
  0,
  0, 0,
  0, 1, 0,
  0, 0, 0, 0,
  0, 1, 3, 1, 0,
  0, 0, 4, 4, 0, 0,
  0, 1, 0, 10, 0, 1, 0,
  0, 0, 6, 4, 4, 6, 0, 0,
  0, 1, 5, 1, 35, 1, 5, 1, 0,
  0, 0, 8, 24, 0, 0, 24, 8, 0, 0,
  0, 1, 0, 4, 84, 126, 84, 4, 0, 1, 0,
  0, 0, 8, 40, 80, 208, 208, 80, 40, 8, 0, 0,
  0, 1, 3, 37, 0, 330, 462, 330, 0, 37, 3, 1, 0,
  0, 0, 0, 64, 204, 264, 792, 792, 264, 204, 64, 0, 0, 0

Cf. A004198, A007318, A285116, A285117.

Cf. A285115 (row sums).

T[n_, k_]:= If[n==0 || n==k, 0, BitAnd[Binomial[n - 1, k - 1], Binomial[n - 1, k]]]; Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 16 2017 *)

T(n, k) = if (n==0 || n==k, 0, bitand(binomial(n - 1, k - 1), binomial(n - 1, k)));
for(n=0, 13, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 16 2017

(define (A285118 n) (A285118tr (A003056 n) (A002262 n)))
(define (A285118tr n k) (cond ((zero? k) 0) ((= k n) 0) (else (A004198bi (A007318tr (- n 1) (- k 1)) (A007318tr (- n 1) k))))) ;; Where A004198bi implements bitwise-AND (A004198) and A007318tr gives the binomial coefficients (A007318).

T(0,n) = T(n,n) = 0; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) AND C(n-1,k), where C(n,k) is binomial coefficient (A007318) & AND is bitwise-AND (A004198).
T(n,k) = A285116(n,k) - A285117(n,k).
A007318(n,k) = C(n,k) = A285116(n,k) + T(n,k) = A285117(n,k) + 2*T(n,k).

A324643 Numbers k such that bitand(2k,sigma(k))/2 = k = bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

Original entry on oeis.org

6, 20, 28, 36, 88, 100, 104, 264, 272, 304, 368, 392, 464, 496, 550, 784, 1032, 1040, 1044, 1056, 1068, 1104, 1120, 1184, 1232, 1312, 1376, 1504, 1696, 1888, 1952, 2140, 3222, 4100, 4128, 4160, 4288, 4512, 4544, 4624, 4640, 4672, 5056, 5312, 5696, 6208, 6328, 6464, 6592, 6808, 6848, 6976, 7232, 7304, 8128, 8288, 8968, 9256, 10184

Offset: 1

Antti Karttunen, Mar 14 2019

Numbers k for which k = A318458(k)/2 = A318468(k).
Intersection of A324649 and A324652.
It is conjectured that there are no odd terms in this sequence, which is equivalent to the conjecture that there are no odd perfect numbers.
Question: Where do the densest clusters of terms occur? See also the scatter plot. - Antti Karttunen, Mar 12 2024
As A324649 and A324652 are both subsequences of nondeficient numbers (A023196), also this sequence is, which stems from the "monotonic property" of bitwise-and. - Antti Karttunen, Jan 08 2025

Antti Karttunen, Table of n, a(n) for n = 1..17020
Michael De Vlieger, Log log scatterplot of a(n), n = 1..17020, labeling cusps in the plot.
Index entries for sequences related to binary expansion of n
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences related to sigma(n)

Cf. A004198, A318458, A318468.
Intersection of A324649 and A324652.
Subsequence of A023196 and of A324639.

Select[Range[10^4], Block[{s = DivisorSigma[1, #]}, # == BitAnd[#, s-#] && 2*# == BitAnd[2*#, s]] &] (* Paolo Xausa, Mar 11 2024 *)

for(n=1,oo,if( (bitand(n, sigma(n)-n)==n) && (bitand(n+n, sigma(n))==2*n),print1(n,", ")))

A324716 a(n) = 2A156552(n) - bitand(2A156552(n), A323243(n)), where bitand is bitwise-AND, A004198.

Original entry on oeis.org

0, 2, 4, 2, 8, 8, 16, 6, 0, 18, 32, 18, 64, 32, 4, 6, 128, 16, 256, 34, 0, 66, 512, 38, 0, 130, 4, 70, 1024, 10, 2048, 30, 64, 258, 0, 22, 4096, 512, 4, 70, 8192, 72, 16384, 130, 8, 1026, 32768, 78, 0, 32, 256, 258, 65536, 32, 0, 134, 4, 2048, 131072, 82, 262144, 4098, 64, 22, 128, 138, 524288, 518, 1024, 80, 1048576, 38, 2097152, 8194, 20, 1030, 0, 266

Offset: 1

Antti Karttunen, Mar 15 2019

nonn

Equivalently, a(n) = 2*A156552(n) XOR (2*A156552(n) AND A323243(n)).

Cf. A003987, A004198, A156552, A323243, A324717, A324722 (positions of zeros).

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552 by David A. Corneth
A324716(n) = { my(t=2*A156552(n)); bitxor(t,bitand(t,A323243(n))); };
\\ Or equivalently:
A324716(n) = { my(t=2*A156552(n)); t - bitand(t,A323243(n)); };

A283974 Numbers n for which A002487(n-1) AND A002487(n) > 0 where AND is bitwise-and (A004198).

Original entry on oeis.org

2, 5, 6, 7, 8, 11, 14, 17, 18, 19, 20, 23, 24, 25, 26, 29, 30, 31, 32, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 86, 89, 92, 95, 96, 97, 98, 101, 104, 107, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120

Offset: 1

Antti Karttunen, Mar 21 2017

Numbers n such that the binary representations of A002487(n-1) and A002487(n) have at least one 1-bit in a common shared position.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A283973 (complement).
Cf. A002487, A004198, A007306, A283986, A283987.
Positions of nonzeros in A283988.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Flatten@ Position[Table[BitAnd[a[n - 1], a@ n], {n, 120}], k_ /; k > 0] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
D(n) = if(n<1, 1, sum(k=0, n, binomial(n + k - 1, 2*k)%2))
for(n=1, 120, if(bitor(A(n - 1), A(n)) != D(n), print1(n, ", "))) \\ Indranil Ghosh, Mar 23 2017

A283998 a(n) = n AND A005187(floor(n/2)), where AND is bitwise-and (A004198).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 4, 4, 0, 1, 8, 8, 8, 8, 10, 11, 0, 1, 16, 16, 16, 16, 18, 19, 16, 16, 18, 19, 24, 25, 26, 26, 0, 1, 32, 32, 32, 32, 34, 35, 32, 32, 34, 35, 40, 41, 42, 42, 32, 32, 34, 35, 48, 49, 50, 50, 48, 49, 50, 50, 56, 56, 56, 57, 0, 1, 64, 64, 64, 64, 66, 67, 64, 64, 66, 67, 72, 73, 74, 74, 64, 64, 66, 67, 80, 81, 82, 82, 80, 81, 82, 82, 88, 88, 88, 89

Offset: 0

Antti Karttunen, Mar 19 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A004198, A005187, A283996, A283997.

A[n_]:=2*n - DigitCount[2*n, 2, 1];Table[BitAnd[n, A[Floor[n/2]]], {n, 0, 100}] (* Indranil Ghosh, Mar 25 2017 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = 2*n - b(2*n);
for(n=0, 100, print1(bitand(n, A(floor(n/2))),", ")) \\ Indranil Ghosh, Mar 25 2017

def A(n): return 2*n - bin(2*n)[2:].count("1")
print([n&A(n//2) for n in range(101)]) # Indranil Ghosh, Mar 25 2017

(define (A283998 n) (A004198bi n (A005187 (floor->exact (/ n 2))))) ;; Where A004198bi implements bitwise-AND (A004198).

a(n) = n AND A005187(floor(n/2)), where AND is bitwise-and (A004198).
a(n) = A283996(n) - A283997(n).
a(n) = A005187(n) - A283996(n) = (A005187(n) - A283997(n))/2.

A318515 a(n) = n AND A032742(n), where AND is bitwise-and (A004198) and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 4, 1, 6, 5, 0, 1, 0, 1, 0, 5, 2, 1, 8, 1, 8, 9, 12, 1, 14, 1, 0, 1, 0, 3, 0, 1, 2, 5, 0, 1, 0, 1, 4, 13, 6, 1, 16, 1, 16, 17, 16, 1, 18, 3, 24, 17, 24, 1, 28, 1, 30, 21, 0, 1, 0, 1, 0, 5, 2, 1, 0, 1, 0, 9, 4, 9, 6, 1, 0, 17, 0, 1, 0, 17, 2, 21, 8, 1, 8, 9, 12, 29, 14, 19, 32, 1, 32, 33, 32, 1, 34, 1, 32, 33

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004198, A032742, A060681, A106409, A318508, A318514, A318518.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318515(n) = bitand(n,A032742(n));

a(n) = A004198(n, A032742(n)).

a(n) = n + A032742(n) - A318514(n) = (n+A032742(n)-A106409(n))/2.

cp's OEIS Frontend

A286099 Square array read by antidiagonals: A(n,k) = T(n OR k, n AND k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and OR is bitwise-or (A003986).

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A318463 a(n) = Sum_{d|n, d < n/d} (d AND n/d), where AND is bitwise-and (A004198).

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A318518 a(n) = A032742(n) AND n-A032742(n), where AND is bitwise-and (A004198) and A032742 = the largest proper divisor of n.

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A283978 a(2n) = 0, a(2n+1) = A002487(n) AND A002487(n+1), where AND is bitwise-and (A004198).

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A285118 Triangle read by rows: T(0,n) = T(n,n) = 0; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) AND C(n-1,k), where C(n,k) is binomial coefficient (A007318) & AND is bitwise-AND (A004198).

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A324643 Numbers k such that bitand(2k,sigma(k))/2 = k = bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

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A324716 a(n) = 2*A156552(n) - bitand(2*A156552(n), A323243(n)), where bitand is bitwise-AND, A004198.

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A283974 Numbers n for which A002487(n-1) AND A002487(n) > 0 where AND is bitwise-and (A004198).

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A324716 a(n) = 2A156552(n) - bitand(2A156552(n), A323243(n)), where bitand is bitwise-AND, A004198.