A115847 -id:A115847 - cp's OEIS frontend

A115848 Sequence A115847 in binary.

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10010, 10100, 10110, 11000, 11010, 11100, 11110, 100000, 100001, 100100, 100101, 101000, 101001, 101100, 101101, 110000, 110100, 111000, 111100

Offset: 0

Antti Karttunen, Feb 01 2006

Cf. a(n) = A007088(A115847(n)). Differs from A007088 (Numbers written in base 2) for the first time at n=17.

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 17, 18, 19, 24, 25, 32, 33, 34, 35, 36, 38, 48, 49, 50, 51, 64, 65, 66, 67, 68, 70, 72, 73, 76, 96, 97, 98, 99, 100, 102, 128, 129, 130, 131, 132, 134, 136, 137, 140, 144, 145, 146, 147, 152, 153, 192, 193, 194, 195, 196, 198, 200, 201

Offset: 1

Antti Karttunen, Mar 30 1999

If bit i is 1, then bits i+-2 must be 0. All terms satisfy A048725(n) = 5*n.
It appears that n is in the sequence if and only if C(5n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
Yes, as remarked in A048715, "This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p." - Jason Kimberley, Dec 21 2011
A116361(a(n)) <= 2. - Reinhard Zumkeller, Feb 04 2006

Superset of A048715 and A048719. Union of A004742 and A003726.

Cf. A048729, A003714, A115845, A115847, A116360.

Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{_, 0, , 1, __}|{_ 1, , 0, __}] && !MatchQ[bb, {_, 1, , 1, __}]];
Select[Range[0, 201], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n,n>>2) \\ Charles R Greathouse IV, Oct 03 2016

list(lim)=my(v=List(),n,t); while(n<=lim, t=bitand(n,n>>2); if(t, n+=1<Charles R Greathouse IV, Oct 22 2021

A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 0, 0, 0, 0, 12, 0, 8, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0

Offset: 0

Antti Karttunen, May 11 2001

			From _Peter Munn_, Jan 28 2021: (Start)
The top left 12 X 12 corner of the table:
      |  0   1   2   3   4   5   6   7   8   9  10  11
------+------------------------------------------------
   0  |  0   0   0   0   0   0   0   0   0   0   0   0
   1  |  0   0   0   0   0   0   0   0   0   0   0   0
   2  |  0   0   0   0   0   0   0   0   0   0   0   0
   3  |  0   0   0   4   0   0   8  12   0   0   0   4
   4  |  0   0   0   0   0   0   0   0   0   0   0   0
   5  |  0   0   0   0   0   8   0   8   0   0  16  16
   6  |  0   0   0   8   0   0  16  24   0   0   0   8
   7  |  0   0   0  12   0   8  24  28   0   0  16  28
   8  |  0   0   0   0   0   0   0   0   0   0   0   0
   9  |  0   0   0   0   0   0   0   0   0  16   0  16
  10  |  0   0   0   0   0  16   0  16   0   0  32  32
  11  |  0   0   0   4   0  16   8  28   0  16  32  52
(End)

"Zoomed in" variant: A061859.
Rows/columns 3, 5 and 7 are given by A048728, A048729, A048730.
Main diagonal divided by 4: A213673.
Numbers that generate no carries when multiplied in binary by 11_2: A003714, by 101_2: A048716, by 1001_2: A115845, by 10001_2: A115847, by 100001_2: A114086.
Other sequences related to the presence/absence of a carry in binary multiplication: A116361, A235034, A235040, A236378, A266195, A289726.

a(n) = A004247(n) - A048720(n).

A116361 Smallest k such that n XOR n2^k = n(2^k + 1).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 4, 2, 4, 3, 4, 1, 1, 1, 2, 1, 1, 4, 5, 2, 2, 4, 5, 3, 5, 4, 5, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 6, 4, 4, 5, 6, 2, 2, 2, 2, 4, 6, 5, 6, 3, 6, 5, 6, 4, 6, 5, 6, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 4, 2, 5, 6, 7, 1, 1, 1, 7, 1, 1, 6, 7, 4, 5, 4, 7, 5, 5, 6, 7, 2, 2, 2, 2, 2, 7, 2, 7, 4

Offset: 0

Reinhard Zumkeller, Feb 04 2006

a(A003714(n)) <= 1;
a(A048716(n)) <= 2;
a(A115845(n)) <= 3;
a(A115847(n)) <= 4;
a(A114086(n)) <= 5;
a(A116362(n)) = n and a(m) < n for m < A116362(n).

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

a[n_] := Module[{k}, For[k = 0, True, k++,
     If[BitXor[n, n*2^k] == n*(2^k+1), Return[k]]]];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 19 2021 *)

a(n)=my(k);while(bitxor(n,n<Charles R Greathouse IV, Mar 07 2013

from itertools import count
def A116361(n): return next(k for k in count(0) if n^(m:=n<Chai Wah Wu, Jul 19 2024

Offset corrected by Charles R Greathouse IV, Mar 07 2013

A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17, 20, 21, 24, 28, 32, 33, 34, 35, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 84, 85, 96, 97, 98, 99, 112, 113, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 140, 142, 160, 161, 162, 163, 168, 170, 192

Offset: 1

Antti Karttunen, Feb 01 2006

nonn

Equivalently, numbers n such that 9*n = 9 X n, i.e., 8*n XOR n = 9*n. Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
Equivalently, numbers n such that the binomial coefficient C(9n,n) (A169958) is odd. - Zak Seidov, Aug 06 2010
The equivalence of these three definitions follows from Lucas's theorem on binomial coefficients. - N. J. A. Sloane, Sep 01 2010
Clearly all numbers k*2^i for 1 <= k <= 7 have this property. - N. J. A. Sloane, Sep 01 2010
A116361(a(n)) <= 3. - Reinhard Zumkeller, Feb 04 2006

N. J. A. Sloane and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences defined by congruent products between domains N and GF(2)[X]
Index entries for sequences defined by congruent products under XOR

A115846 shows this sequence in binary.
A033052 is a subsequence.
Cf. A003714, A048716, A115847, A116360, A005809, A003714, A048716, A048715.

Reap[Do[If[OddQ[Binomial[9n,n]],Sow[n]],{n,0,400}]][[2,1]] (* Zak Seidov, Aug 06 2010 *)

is(n)=!bitand(n,n<<3) \\ Charles R Greathouse IV, Sep 23 2012

a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 23 2012

Edited with a new definition by N. J. A. Sloane, Sep 01 2010, merging this sequence with a sequence submitted by Zak Seidov, Aug 06 2010

A114086 Numbers m such that m XOR 32m = 33m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 100, 104, 108, 112, 116, 120

Offset: 0

Reinhard Zumkeller, Feb 04 2006

nonn

A116361(a(n)) <= 5.

Index entries for sequences defined by congruent products under XOR

Differs from A001477 for the first time at n=33 (33, 35, 37, 39, etc. are not present in this sequence). Cf. A003714, A048716, A115845, A115847.

is(n)=bitxor(n, 32*n)==33*n \\ Charles R Greathouse IV, Sep 25 2024

A115849 Integers i such that 17*i is not 17 X i.

Original entry on oeis.org

17, 19, 21, 23, 25, 27, 29, 31, 34, 35, 38, 39, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 68, 69, 70, 71, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 98, 99, 100, 101, 102, 103, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116

Offset: 1

Antti Karttunen, Feb 01 2006

nonn

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Cf. A115850 shows this sequence in binary. Complement of A115847.

cp's OEIS Frontend

A115848 Sequence A115847 in binary.

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A048716 Numbers n such that binary expansion matches ((0)*00(1?)1)*(0*).

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A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

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A116361 Smallest k such that n XOR n*2^k = n*(2^k + 1).

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A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

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A114086 Numbers m such that m XOR 32*m = 33*m.

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PARI

A115849 Integers i such that 17*i is not 17 X i.

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A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

A116361 Smallest k such that n XOR n2^k = n(2^k + 1).

A114086 Numbers m such that m XOR 32m = 33m.