A003986 -id:A003986 - cp's OEIS frontend

A318456 a(n) = n OR A001065(n), where OR is bitwise-or (A003986) and A001065 = sum of proper divisors.

1, 3, 3, 7, 5, 6, 7, 15, 13, 10, 11, 28, 13, 14, 15, 31, 17, 23, 19, 22, 31, 30, 23, 60, 31, 26, 31, 28, 29, 62, 31, 63, 47, 54, 47, 55, 37, 54, 55, 58, 41, 62, 43, 44, 45, 62, 47, 124, 57, 59, 55, 62, 53, 118, 55, 120, 63, 58, 59, 124, 61, 62, 63, 127, 83, 78, 67, 126, 95, 78, 71, 123, 73, 106, 123, 76, 95, 94, 79, 122, 121, 126, 83

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A001065, A003986, A318457, A318458, A318466.

Array[BitOr[#, DivisorSigma[1, #] - #] &, 100] (* Paolo Xausa, Mar 11 2024 *)

PARI
```
A318456(n) = bitor(n,sigma(n)-n);
```

from sympy import divisor_sigma
def A318456(n): return n|divisor_sigma(n)-n # Chai Wah Wu, Jul 01 2022

a(n) = A003986(n, A001065(n)).

a(n) = A000203(n) - A318458(n).

A324866 a(n) = A156552(n) OR A324865(n), where OR is bitwise-OR, A003986.

Original entry on oeis.org

0, 1, 3, 3, 7, 5, 15, 7, 6, 13, 31, 11, 63, 17, 10, 15, 127, 13, 255, 19, 23, 47, 511, 23, 28, 83, 14, 47, 1023, 31, 2047, 31, 54, 175, 22, 31, 4095, 257, 78, 55, 8191, 37, 16383, 67, 30, 799, 32767, 47, 60, 31, 250, 131, 65535, 29, 55, 71, 270, 1301, 131071, 43, 262143, 2735, 54, 63, 126, 95, 524287, 303, 774, 41, 1048575, 55

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. A001065, A003986, A156552, A318456, A323243, A324865, A324867, A324398, A324876.

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
A318456(n) = bitor(n,sigma(n)-n);
A324866(n) = if(1==n,0,A318456(A156552(n)));

A324866(n) = { my(k=A156552(n)); bitor(k,(A323243(n)-k)); }; \\ Needs also code from A323243.

a(1) = 0; for n > 1, a(n) = A318456(A156552(n)).

a(n) = A156552(n) OR (A323243(n) - A156552(n)).

A318466 a(n) = 2*n OR A000203(n), where OR is bitwise-or (A003986) and A000203 = sum of divisors.

Original entry on oeis.org

3, 7, 6, 15, 14, 12, 14, 31, 31, 22, 30, 28, 30, 28, 30, 63, 50, 39, 54, 42, 42, 44, 62, 60, 63, 62, 62, 56, 62, 124, 62, 127, 114, 118, 118, 91, 110, 124, 126, 90, 122, 116, 126, 92, 94, 92, 126, 124, 123, 125, 110, 106, 126, 124, 110, 120, 114, 126, 126, 248, 126, 124, 126, 255, 214, 148, 198, 254, 234, 156, 206

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003986, A224880, A318467, A318468.

Cf. also A318456.

Array[BitOr[2 #, DivisorSigma[1, #]] &, 71] (* Michael De Vlieger, Mar 30 2019 *)

PARI
```
A318466(n) = bitor(2*n,sigma(n));
```

from sympy import divisor_sigma
def A318466(n): return (n<<1)|int(divisor_sigma(n)) # Chai Wah Wu, Jul 10 2022

a(n) = A003986(2*n, A000203(n)).

a(n) = A224880(n) - A318468(n).

A283986 a(n) = A002487(n-1) OR A002487(n), where OR is bitwise-or (A003986).

Original entry on oeis.org

1, 1, 3, 3, 3, 3, 3, 3, 5, 7, 7, 7, 7, 7, 7, 5, 5, 5, 7, 7, 11, 13, 7, 7, 7, 7, 13, 11, 7, 7, 5, 5, 7, 7, 13, 13, 15, 15, 15, 11, 11, 11, 13, 13, 13, 15, 15, 11, 11, 15, 15, 13, 13, 13, 11, 11, 11, 15, 15, 15, 13, 13, 7, 7, 7, 7, 15, 15, 15, 15, 13, 13, 15, 15, 27, 23, 23, 27, 15, 15, 15, 15, 27, 27, 29, 29, 31, 23, 21, 29, 31, 23, 23, 25, 11, 11, 11, 11, 25

Offset: 1

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Odd bisection of A283976.
Cf. A002487, A003986, A283988.
Cf. A283973 (positions where coincides with A007306, equally, with A283987).

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[BitOr[a[n - 1], a@ n], {n, 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)));
for(n=1, 101, print1(bitor(A(n - 1), A(n))", ")) \\ Indranil Ghosh, Mar 23 2017

from functools import reduce
def A283986(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0)))|sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n-1)[-1:2:-1],(1,0))) # Chai Wah Wu, May 05 2023

(define (A283986 n) (A003986bi (A002487 (- n 1)) (A002487 n))) ;; Where A003986bi implements bitwise-OR (A003986).

a(n) = A002487(n-1) OR A002487(n), where OR is bitwise-or (A003986).
a(n) = A283987(n) + A283988(n).
a(n) = A007306(n) - A283988(n).
a(n) = A283976((2*n)-1).

A324727 Odd numbers such that 2n is equal to A318466(n), bitor(2*n,sigma(n)), where bitor is A003986.

Original entry on oeis.org

3, 7, 15, 21, 31, 55, 57, 63, 93, 105, 111, 127, 171, 189, 201, 213, 215, 217, 231, 237, 249, 253, 255, 315, 351, 357, 363, 369, 381, 393, 447, 465, 469, 473, 483, 489, 497, 501, 511, 651, 705, 747, 759, 789, 813, 831, 833, 879, 889, 895, 917, 959, 987, 989, 1001, 1015, 1023, 1155, 1365, 1377, 1407, 1467, 1491, 1503, 1505, 1515, 1533, 1595

Offset: 1

Antti Karttunen, Mar 15 2019

Odd terms of A324726.

Cf. A000203, A003986, A318466, A324647, A324718, A324719.

Select[Range[1, 2000, 2], 2*# == BitOr[2*#, DivisorSigma[1, #]] &] (* Paolo Xausa, Mar 11 2024 *)

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

A318506 a(n) = A032742(n) OR A001065(n)-A032742(n), where OR is bitwise-or (A003986) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 7, 3, 7, 1, 14, 1, 7, 5, 15, 1, 13, 1, 14, 7, 11, 1, 28, 5, 15, 13, 14, 1, 31, 1, 31, 15, 19, 7, 55, 1, 19, 13, 30, 1, 53, 1, 22, 31, 23, 1, 60, 7, 27, 21, 30, 1, 63, 15, 60, 23, 31, 1, 94, 1, 31, 21, 63, 15, 45, 1, 58, 23, 39, 1, 119, 1, 39, 25, 62, 11, 55, 1, 106, 31, 43, 1, 106, 23, 43, 29, 60, 1, 111, 13, 62, 31

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. A001065, A003986, A032742, A318505, A318507, A318508.

Cf. also A318456, A318516.

A001065(n) = (sigma(n)-n);
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318505(n) = if(1==n,0,A001065(n)-A032742(n));
A318506(n) = bitor(A318505(n),A032742(n));

a(n) = A003986(A032742(n), A318505(n)).

For n > 1, a(n) = A001065(n) - A318508(n).

A286098 Square array read by antidiagonals: A(n,k) = T(n AND k, n OR 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).

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 6, 6, 6, 6, 10, 11, 12, 11, 10, 15, 15, 17, 17, 15, 15, 21, 22, 21, 24, 21, 22, 21, 28, 28, 28, 28, 28, 28, 28, 28, 36, 37, 38, 37, 40, 37, 38, 37, 36, 45, 45, 47, 47, 49, 49, 47, 47, 45, 45, 55, 56, 55, 58, 59, 60, 59, 58, 55, 56, 55, 66, 66, 66, 66, 70, 70, 70, 70, 66, 66, 66, 66, 78, 79, 80, 79, 78, 83, 84, 83, 78, 79, 80, 79, 78

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,  1,   3,   6,  10,  15,  21,  28,  36,  45,  55,  66,  78
   1,  4,   6,  11,  15,  22,  28,  37,  45,  56,  66,  79,  91
   3,  6,  12,  17,  21,  28,  38,  47,  55,  66,  80,  93, 105
   6, 11,  17,  24,  28,  37,  47,  58,  66,  79,  93, 108, 120
  10, 15,  21,  28,  40,  49,  59,  70,  78,  91, 105, 120, 140
  15, 22,  28,  37,  49,  60,  70,  83,  91, 106, 120, 137, 157
  21, 28,  38,  47,  59,  70,  84,  97, 105, 120, 138, 155, 175
  28, 37,  47,  58,  70,  83,  97, 112, 120, 137, 155, 174, 194
  36, 45,  55,  66,  78,  91, 105, 120, 144, 161, 179, 198, 218
  45, 56,  66,  79,  91, 106, 120, 137, 161, 180, 198, 219, 239
  55, 66,  80,  93, 105, 120, 138, 155, 179, 198, 220, 241, 261
  66, 79,  93, 108, 120, 137, 155, 174, 198, 219, 241, 264, 284
  78, 91, 105, 120, 140, 157, 175, 194, 218, 239, 261, 284, 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. A000217 (row 0 & column 0), A084263 (seems to be row 1 & column 1), A046092 (main diagonal).
Cf. A003056, A003986, A004198.
Cf. also arrays A286099, A286101, A286102, A286108.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitAnd[n, k],BitOr[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 (A286098 n) (A286098bi (A002262 n) (A025581 n)))
(define (A286098bi row col) (let ((a (A004198bi row col)) (b (A003986bi 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(A004198(n,k), A003986(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, ...].

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).

Original entry on oeis.org

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, ...].

A318516 a(n) = A032742(n) OR n-A032742(n), where OR is bitwise-or (A003986) and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 1, 3, 2, 5, 3, 7, 4, 7, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 15, 11, 23, 12, 21, 13, 27, 14, 29, 15, 31, 16, 31, 17, 31, 18, 37, 19, 31, 20, 41, 21, 43, 22, 31, 23, 47, 24, 47, 25, 51, 26, 53, 27, 47, 28, 55, 29, 59, 30, 61, 31, 63, 32, 61, 33, 67, 34, 63, 35, 71, 36, 73, 37, 59, 38, 75, 39, 79, 40, 63, 41, 83, 42

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. A003986, A032742, A060681, A318506, A318514, A318517, A318518.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318516(n) = bitor(A032742(n),n-A032742(n));

a(n) = A003986(A032742(n), A060681(n)).

a(n) = n - A318518(n).

A324719 Odd numbers n for which bitor(2n,sigma(n)) = 2*bitor(n,sigma(n)-n), where bitor is bitwise-OR, A003986.

Original entry on oeis.org

3, 7, 15, 27, 31, 51, 55, 63, 111, 119, 123, 125, 127, 219, 255, 411, 447, 485, 493, 495, 505, 511, 735, 765, 771, 831, 879, 927, 959, 965, 985, 1011, 1023, 1563, 1587, 1611, 1731, 1779, 1791, 1799, 1887, 1921, 1923, 1945, 1975, 1983, 1991, 2019, 2031, 2041, 2043, 2045, 2047, 3099, 3183, 3231, 3279, 3291, 3327, 3459, 3535, 3579

Offset: 1

Antti Karttunen, Mar 14 2019

Odd numbers n for which 2*A318456(n) = A318466(n).
If there are no common terms with A324718, then there are no odd perfect numbers.
The following subsequence of terms k are those with sigma(k) == 2 (mod 4): 3725, 7281, 15325, 24525, 25713, 32481, 51633, 52209, 59121, 63553, 114417, 117009, 120753, 121725, 122725, 123245, 130833, 208881, 236925, 241325, 245725, 253325, 261297, 384993, 411633, 457713, 468081, 482481, 482525, 482725, 483325, ..., and are thus present in A191218.

Cf. A003986, A191218, A318456, A318466, A324718, A324727.

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

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

cp's OEIS Frontend

A318456 a(n) = n OR A001065(n), where OR is bitwise-or (A003986) and A001065 = sum of proper divisors.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A324866 a(n) = A156552(n) OR A324865(n), where OR is bitwise-OR, A003986.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A318466 a(n) = 2*n OR A000203(n), where OR is bitwise-or (A003986) and A000203 = sum of divisors.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A283986 a(n) = A002487(n-1) OR A002487(n), where OR is bitwise-or (A003986).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A324727 Odd numbers such that 2n is equal to A318466(n), bitor(2*n,sigma(n)), where bitor is A003986.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A318506 a(n) = A032742(n) OR A001065(n)-A032742(n), where OR is bitwise-or (A003986) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A286098 Square array read by antidiagonals: A(n,k) = T(n AND k, n OR 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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

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).

Views

Author

Keywords