A003986 -id:A003986 - cp's OEIS frontend

A324819 a(n) = 2*A156552(n) OR A323243(n), where OR is bitwise-OR, A003986.

0, 3, 7, 6, 15, 14, 31, 14, 12, 31, 63, 30, 127, 50, 22, 30, 255, 30, 511, 54, 39, 114, 1023, 62, 28, 214, 28, 118, 2047, 42, 4095, 62, 118, 434, 42, 62, 8191, 770, 148, 126, 16383, 110, 32767, 198, 44, 1826, 65535, 126, 60, 63, 508, 390, 131071, 62, 91, 206, 532, 3350, 262143, 126, 524287, 6834, 124, 126, 254, 234, 1048575, 822

Offset: 1

Antti Karttunen, Mar 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003986, A156552, A323243, A324815, A324821.

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
A324819(n) = bitor(2*A156552(n),A323243(n)); \\ Needs code also from A323243.

a(A000040(n)) = A000225(1+n).

A324821 Xor-Moebius transform of A324819, where A324819(n) = bitor(2*A156552(n), A323243(n)), where bitor is A003986.

Original entry on oeis.org

0, 3, 7, 5, 15, 10, 31, 8, 11, 19, 63, 21, 127, 46, 30, 16, 255, 27, 511, 44, 63, 78, 1023, 40, 19, 170, 16, 65, 2047, 38, 4095, 32, 78, 334, 58, 48, 8191, 766, 236, 64, 16383, 110, 32767, 177, 49, 1246, 65535, 80, 35, 51, 260, 341, 131071, 48, 107, 176, 1004, 2794, 262143, 104, 524287, 5454, 80, 64, 142, 219, 1048575, 641, 1278, 110

Offset: 1

Antti Karttunen, Mar 18 2019

nonn

It seems that the records, which are A000225(1+n) = 2^(1+n) - 1 occur at primes, as occur also the records for the width of terms, A000523(a(n)), and the records for the binary weight of terms, A000120(a(n)).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003986, A156552, A323243, A324815, A324819, A324820, A324876, A324878.

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
A324819(n) = bitor(2*A156552(n),A323243(n)); \\ Needs code also from A323243.
A324821(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A324819(d)))); (v); };

a(A000040(n)) = A000225(1+n).

A283976 a(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 3, 1, 3, 3, 3, 2, 3, 3, 3, 1, 5, 4, 7, 3, 7, 5, 7, 2, 7, 5, 7, 3, 7, 4, 5, 1, 5, 5, 5, 4, 7, 7, 7, 3, 11, 8, 13, 5, 7, 7, 7, 2, 7, 7, 7, 5, 13, 8, 11, 3, 7, 7, 7, 4, 5, 5, 5, 1, 7, 6, 7, 5, 13, 9, 13, 4, 15, 11, 15, 7, 15, 10, 11, 3, 11, 11, 11, 8, 13, 13, 13, 5, 13, 12, 15, 7, 15, 9, 11, 2, 11, 9, 15, 7, 15, 12, 13, 5, 13, 13, 13, 8

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: A002487, A283986.

Cf. A002487, A003986, A283977, A283978.

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, a[n/2], BitOr[a[#], a[# + 1]] &[(n - 1)/2]], {n, 0, 108}] (* 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, n, if(n%2, bitor(A(n\2), A((n + 1)/2)), A(n\2)));
for(n=0, 101, print1(a(n),", ")) \\ Indranil Ghosh, Mar 23 2017

(define (A283976 n) (if (even? n) (A002487 n) (A003986bi (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))) ;; Where A003986bi implements bitwise-OR (A003986).

a(2n) = A002487(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).
a(n) = A283977(n) + A283978(n).
a(n) = A002487(n) - A283978(n).

A285116 Triangle read by rows: T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) OR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and OR is bitwise-OR (A003986).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 3, 3, 3, 1, 1, 5, 6, 6, 5, 1, 1, 5, 15, 10, 15, 5, 1, 1, 7, 15, 31, 31, 15, 7, 1, 1, 7, 23, 55, 35, 55, 23, 7, 1, 1, 9, 28, 60, 126, 126, 60, 28, 9, 1, 1, 9, 45, 116, 126, 126, 126, 116, 45, 9, 1, 1, 11, 47, 125, 250, 254, 254, 250, 125, 47, 11, 1, 1, 11, 63, 183, 495, 462, 462, 462, 495, 183, 63, 11, 1

Offset: 0

Antti Karttunen, Apr 16 2017

			Rows 0 - 12 of the triangle:
  1,
  1, 1,
  1, 1, 1,
  1, 3, 3, 1,
  1, 3, 3, 3, 1,
  1, 5, 6, 6, 5, 1,
  1, 5, 15, 10, 15, 5, 1,
  1, 7, 15, 31, 31, 15, 7, 1,
  1, 7, 23, 55, 35, 55, 23, 7, 1,
  1, 9, 28, 60, 126, 126, 60, 28, 9, 1,
  1, 9, 45, 116, 126, 126, 126, 116, 45, 9, 1,
  1, 11, 47, 125, 250, 254, 254, 250, 125, 47, 11, 1,
  1, 11, 63, 183, 495, 462, 462, 462, 495, 183, 63, 11, 1

Cf. A003986, A007318, A285117, A285118.

Cf. A285113 (row sums).

T[n_, k_]:= If[n==0 || n==k, 1, BitOr[Binomial[n - 1, k - 1], Binomial[n - 1, k]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 16 2017 *)

T(n, k) = if (n==0 || n==k, 1, bitor(binomial(n - 1, k - 1), binomial(n - 1, k)));
for(n=0, 12, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 16 2017

(define (A285116 n) (A285116tr (A003056 n) (A002262 n)))
(define (A285116tr n k) (cond ((zero? k) 1) ((= k n) 1) (else (A003986tr (A007318tr (- n 1) (- k 1)) (A007318tr (- n 1) k))))) ;; Where A003986bi implements bitwise-OR (A003986) and A007318tr gives the binomial coefficients (A007318).

T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) OR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and OR is bitwise-OR (A003986).
T(n,k) = A285117(n,k) + A285118(n,k).
C(n,k) = T(n,k) + A285118(n,k). [Where C(n,k) = A007318.]

A318461 a(n) = Sum_{d|n, d <= n/d} (d OR n/d), where OR is bitwise-or (A003986).

Original entry on oeis.org

1, 3, 3, 7, 5, 10, 7, 15, 12, 18, 11, 26, 13, 22, 22, 31, 17, 37, 19, 36, 28, 34, 23, 56, 30, 42, 38, 50, 29, 64, 31, 63, 44, 54, 42, 89, 37, 58, 54, 90, 41, 88, 43, 82, 73, 70, 47, 120, 56, 93, 70, 92, 53, 116, 70, 116, 76, 90, 59, 156, 61, 94, 101, 127, 78, 140, 67, 124, 92, 136, 71, 183, 73, 114, 117, 138, 92, 160, 79

Offset: 1

Antti Karttunen, Aug 28 2018

Cf. A000203, A003986, A318456, A318462, A318463.

A318461(n) = { my(ors=0); fordiv(n,d,if(d<=(n/d), ors += bitor(d,n/d))); (ors); };

a(n) = A000203(n) - A318463(n).

A324726 Numbers k such that 2k is equal to 2k OR sigma(k), where OR is bitwise-or, A003986, and sigma is the sum of divisors function.

Original entry on oeis.org

3, 6, 7, 14, 15, 21, 22, 28, 31, 46, 55, 57, 62, 63, 86, 92, 93, 94, 105, 110, 111, 124, 127, 154, 170, 171, 188, 189, 190, 201, 213, 215, 217, 231, 237, 248, 249, 250, 253, 254, 255, 310, 315, 316, 351, 357, 363, 369, 374, 376, 381, 382, 393, 430, 434, 447, 465, 469, 473, 483, 489, 494, 496, 497, 501, 506, 508, 511, 602

Offset: 1

Antti Karttunen, Mar 15 2019

Intersection with A324652 gives A000396.

These are all nonabundant (in A263837) because of the "monotonic property" of bitwise-or. - Antti Karttunen, Jan 08 2025

Cf. A000396, A003986, A318466, A324652, A324723, A324727 (the odd terms).

Subsequence of A263837.

Select[Range[1000], 2*# == BitOr[2*#, DivisorSigma[1, #]] &] (* Paolo Xausa, Mar 11 2024 *)

for(n=1, oo, if(((2*n)==bitor(2*n, sigma(n))), print1(n, ", ")));

{k such that 2*k = A318466(k)}.

A350093 a(n) = Sum_{k=0..n} n OR k where OR is the bitwise logical OR operator (A003986).

Original entry on oeis.org

0, 2, 7, 12, 26, 34, 45, 56, 100, 114, 131, 148, 174, 194, 217, 240, 392, 418, 447, 476, 514, 546, 581, 616, 684, 722, 763, 804, 854, 898, 945, 992, 1552, 1602, 1655, 1708, 1770, 1826, 1885, 1944, 2036, 2098, 2163, 2228, 2302, 2370, 2441, 2512, 2712, 2786, 2863

Offset: 0

Kevin Ryde, Dec 14 2021

The effect of n OR k is to force a 1-bit at all bit positions where n has a 1-bit, which means n*(n+1) in the sum. Bits of k where n has a 0-bit are NOT(n) AND k = n CNIMPL k so that a(n) = A350094(n) + n*(n+1).

Winston de Greef, Table of n, a(n) for n = 0..10000

Cf. A003986 (bitwise OR), A001196 (bit doubling).
Row sums of A080098.
Other sums: A222423 (AND), A224915 (XOR), A265736 (IMPL), A350094 (CNIMPL).

a(n) = (3*(n^2 + fromdigits(binary(n),4)) + 2*n) >> 2;

a(n) = ((3*n+2)*n + A001196(n)) / 4.
a(2*n) = 4*a(n) - n.
a(2*n+1) = 4*a(n) + 2*n + 2.
a(n) = A222423(n) + A224915(n), being OR = AND + XOR.

A276010 a(0) = 0, for n >= 1, a(n) = A275736(n) OR a(A257684(n)), where OR is given by A003986.

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 4, 5, 6, 7, 5, 5, 2, 3, 2, 3, 3, 3, 1, 1, 3, 3, 1, 1, 8, 9, 10, 11, 9, 9, 12, 13, 14, 15, 13, 13, 10, 11, 10, 11, 11, 11, 9, 9, 11, 11, 9, 9, 4, 5, 6, 7, 5, 5, 4, 5, 6, 7, 5, 5, 6, 7, 6, 7, 7, 7, 5, 5, 7, 7, 5, 5, 2, 3, 2, 3, 3, 3, 6, 7, 6, 7, 7, 7, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 3, 3, 1, 1, 5, 5, 7, 7, 5, 5, 3

Offset: 0

Antti Karttunen, Aug 17 2016

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A003986, A087207, A257684, A275734.

Cf. also A000120, A060502, A275736, A275728, A275808.

a(0) = 0, for n >= 1, a(n) = A275736(n) OR a(A257684(n)), where OR is given by A003986.
a(n) = A087207(A275734(n)).
Other identities. For all n >= 1:
A000120(a(n)) = A060502(n).

A283996 a(n) = n OR A005187(floor(n/2)), where OR is bitwise-or (A003986).

Original entry on oeis.org

0, 1, 3, 3, 7, 7, 6, 7, 15, 15, 10, 11, 14, 15, 15, 15, 31, 31, 18, 19, 22, 23, 23, 23, 30, 31, 31, 31, 29, 29, 30, 31, 63, 63, 34, 35, 38, 39, 39, 39, 46, 47, 47, 47, 45, 45, 46, 47, 62, 63, 63, 63, 53, 53, 54, 55, 61, 61, 62, 63, 60, 61, 63, 63, 127, 127, 66, 67, 70, 71, 71, 71, 78, 79, 79, 79, 77, 77, 78, 79, 94, 95, 95, 95, 85, 85, 86, 87, 93, 93, 94, 95

Offset: 0

Antti Karttunen, Mar 19 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A003986, A005187, A283997, A283998.

A[n_]:=2*n - DigitCount[2*n, 2, 1]; Table[BitOr[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(bitor(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 (A283996 n) (A003986bi n (A005187 (floor->exact (/ n 2))))) ;; Where A003986bi implements bitwise-OR (A003986).

a(n) = n OR A005187(floor(n/2)), where OR is bitwise-or (A003986).
a(n) = A283997(n) + A283998(n).
a(n) = A005187(n) - A283998(n).

A325316 a(n) = A048250(n) OR A162296(n), where OR is the bitwise-OR, A003986.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 31, 20, 26, 32, 36, 24, 60, 31, 42, 36, 56, 30, 72, 32, 63, 48, 54, 48, 79, 38, 60, 56, 90, 42, 96, 44, 52, 62, 72, 48, 124, 57, 91, 72, 58, 54, 108, 72, 120, 80, 90, 60, 104, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 191, 74, 114, 124, 124, 96, 168, 80

Offset: 1

Antti Karttunen, Apr 21 2019

nonn

Cf. A000203, A003986, A048250, A162296, A325317, A325318.

Array[BitOr @@ Map[Total, {#3, Complement[#2, #3]}] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 79] (* Michael De Vlieger, Apr 21 2019 *)

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325316(n) = bitor(A048250(n),A162296(n));

a(n) = A003986(A048250(n), A162296(n)).

a(n) = A000203(n) - A325318(n) = A325317(n) + A325318(n).

cp's OEIS Frontend

A324819 a(n) = 2*A156552(n) OR A323243(n), where OR is bitwise-OR, A003986.

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A324821 Xor-Moebius transform of A324819, where A324819(n) = bitor(2*A156552(n), A323243(n)), where bitor is A003986.

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A283976 a(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).

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A285116 Triangle read by rows: T(0,n) = T(n,n) = 1; and for n > 0, 0 < k < n, T(n,k) = C(n-1,k-1) OR C(n-1,k), where C(n,k) is binomial coefficient (A007318) and OR is bitwise-OR (A003986).

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A318461 a(n) = Sum_{d|n, d <= n/d} (d OR n/d), where OR is bitwise-or (A003986).

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A324726 Numbers k such that 2*k is equal to 2*k OR sigma(k), where OR is bitwise-or, A003986, and sigma is the sum of divisors function.

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A350093 a(n) = Sum_{k=0..n} n OR k where OR is the bitwise logical OR operator (A003986).

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A276010 a(0) = 0, for n >= 1, a(n) = A275736(n) OR a(A257684(n)), where OR is given by A003986.

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A324726 Numbers k such that 2k is equal to 2k OR sigma(k), where OR is bitwise-or, A003986, and sigma is the sum of divisors function.