A115845 -id:A115845

Search: a115845 -id:a115845

Displaying 1-6 of 6 results found.

page 1

A115846

Sequence A115845 in binary.

+20
1

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1010, 1100, 1110, 10000, 10001, 10100, 10101, 11000, 11100, 100000, 100001, 100010, 100011, 101000, 101010, 110000, 110001, 111000, 1000000, 1000001, 1000010, 1000011, 1000100, 1000101

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..32.

CROSSREFS

Cf. a(n) = A007088(A115845(n)).

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Feb 01 2006

STATUS

approved

A048716

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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

Index entries for 2-automatic sequences.

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

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

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

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

KEYWORD

nonn,base,easy

AUTHOR

Antti Karttunen, Mar 30 1999

STATUS

approved

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.

+10
12

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,25

LINKS

Table of n, a(n) for n=0..101.

FORMULA

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

EXAMPLE

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)

CROSSREFS

"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.

KEYWORD

nonn,tabl,easy

AUTHOR

Antti Karttunen, May 11 2001

STATUS

approved

A116361

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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).

LINKS

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

MATHEMATICA

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 *)

PROG

(PARI) a(n)=my(k); while(bitxor(n, n<<k)-n!=n<<k, k++); k \\ Charles R Greathouse IV, Mar 07 2013

(Python)

from itertools import count

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

KEYWORD

nonn,easy

AUTHOR

Reinhard Zumkeller, Feb 04 2006

EXTENSIONS

Offset corrected by Charles R Greathouse IV, Mar 07 2013

STATUS

approved

A115847

Integers i such that 17*i = 17 X i, i.e., 16*i XOR i = 17*i.

+10
8

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 36, 37, 40, 41, 44, 45, 48, 52, 56, 60, 64, 65, 66, 67, 72, 73, 74, 75, 80, 82, 88, 90, 96, 97, 104, 105, 112, 120, 128, 129, 130, 131, 132, 133, 134, 135, 144, 146, 148, 150

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

A116361(a(n)) <= 4. - Reinhard Zumkeller, Feb 04 2006

LINKS

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

PROG

(PARI) is(n)=bitxor(n, 16*n)==17*n \\ Charles R Greathouse IV, Dec 08 2013

CROSSREFS

Cf. A115848 shows this sequence in binary. Complement of A115849. Differs from A032966 for the first time at n=25, where A032966(25)=34 while a(25)=33.

Cf. A003714, A048716, A115845, A116360.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 01 2006

STATUS

approved

A114086

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

A116361(a(n)) <= 5.

LINKS

Table of n, a(n) for n=0..70.

Index entries for sequences defined by congruent products under XOR

PROG

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

CROSSREFS

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.