A080079 -id:A080079 - cp's OEIS frontend

A327489 T(n, k) = 1 + NOR(k - 1, n - k), where NOR is the Peirce arrow operating bitwise on the inputs, triangle read by rows, T(n, k) for n >= 1, 1 <= k <= n.

1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 1, 4, 3, 3, 1, 1, 3, 3, 2, 3, 2, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 7, 7, 1, 1, 3, 3, 1, 1, 7, 7, 6, 7, 6, 1, 2, 3, 2, 1, 6, 7, 6, 5, 5, 5, 5, 1, 1, 1, 1, 5, 5, 5, 5

Offset: 1

Peter Luschny, Sep 22 2019

			                               1
                              1, 1
                            2, 1, 2
                           1, 1, 1, 1
                         4, 1, 2, 1, 4
                        3, 3, 1, 1, 3, 3
                      2, 3, 2, 1, 2, 3, 2
                     1, 1, 1, 1, 1, 1, 1, 1
                   8, 1, 2, 1, 4, 1, 2, 1, 8
                  7, 7, 1, 1, 3, 3, 1, 1, 7, 7
                6, 7, 6, 1, 2, 3, 2, 1, 6, 7, 6
               5, 5, 5, 5, 1, 1, 1, 1, 5, 5, 5, 5
             4, 5, 6, 5, 4, 1, 2, 1, 4, 5, 6, 5, 4
            3, 3, 5, 5, 3, 3, 1, 1, 3, 3, 5, 5, 3, 3
          2, 3, 2, 5, 2, 3, 2, 1, 2, 3, 2, 5, 2, 3, 2
         1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Cf. A327488 (Nand), A327490 (Iff), A280172 (Xor).

T(2n+1,n+1) gives A080079.

A327489 := (n, k) -> 1 + Bits:-Nor(k-1, n-k):
seq(seq(A327489(n, k), k=1..n), n=1..12);

A097327 Least positive integer m such that m*n has greater decimal digit length than n.

Original entry on oeis.org

10, 5, 4, 3, 2, 2, 2, 2, 2, 10, 10, 9, 8, 8, 7, 7, 6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 10

Offset: 1

Rick L. Shepherd, Aug 04 2004

For any positive base B >= 2 the corresponding sequence contains only terms from 2 to B inclusive so the corresponding sequence for binary is all 2s (A007395).

			a(12) = 9 since 12 has two decimal digits and 9*12 = 108 has three (but 8*12 = 96 has only two).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A089186 (analog for decimal m+n), A080079 (analog for binary m+n), A097326.

Cf. A055642.

Table[Ceiling[10^IntegerLength[n]/n], {n, 100}] (* Paolo Xausa, Nov 02 2024 *)

a(n) = my(m=1, sn=#Str(n)); while (#Str(m*n) <= sn, m++); m; \\ Michel Marcus, Oct 05 2021

def a(n): return (10**len(str(n))-1)//n + 1
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Oct 05 2021

a(n) = A097326(n) + 1.

a(n) = ceiling(10^A055642(n)/n). - Michael S. Branicky, Oct 05 2021

A101267 a(1) = 1; a(n) = a(2^ceiling(log_2(n)) + 1 - n)th smallest positive integer not yet in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 7, 9, 6, 5, 13, 15, 19, 17, 11, 14, 10, 8, 24, 27, 32, 29, 37, 40, 35, 33, 21, 23, 30, 26, 18, 22, 16, 12, 44, 49, 56, 52, 62, 67, 59, 57, 73, 76, 82, 79, 69, 74, 66, 63, 39, 43, 50, 46, 58, 64, 54, 51, 34, 38, 47, 42, 28, 36, 25, 20, 84, 90, 102, 94, 110, 116, 106

Offset: 1

Leroy Quet, Dec 18 2004

Sequence is a permutation of the positive integers. 2^ceiling(log_2(n)) + 1 - n is sequence A080079 with a change of offset.

			Since 2^ceiling(log_2(n)) +1 -n = 3 at n = 6, a(6) = the a(3)th (the 4th) smallest positive integer not among the first 5 terms of the sequence. The positive integers not among the first 5 terms are 5,6,8,9,10,... The 4th of these is 9, which is a(6).

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A101283, A119474.

a[1] = 1; a[n_] := a[n] = Complement[ Range[100], Table[ a[i], {i, n - 1}]] [[ a[2^Ceiling[ Log[2, n]] + 1 - n]]]; Table[ a[n], {n, 71}] (* Robert G. Wilson v, Jan 13 2005 *)

More terms from Robert G. Wilson v, Jan 13 2005

A240769 Triangle read by rows: T(1,1) = 1; T(n+1,k) = T(n,k+1), 1 <= k < n; T(n+1,n) = 2T(n,1); T(n+1,n+1) = 2T(n,1) - 1.

Original entry on oeis.org

1, 2, 1, 1, 4, 3, 4, 3, 2, 1, 3, 2, 1, 8, 7, 2, 1, 8, 7, 6, 5, 1, 8, 7, 6, 5, 4, 3, 8, 7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 16, 15, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 3, 2

Offset: 1

Reinhard Zumkeller, Apr 13 2014

Let h be the initial term of row n, to get row n+1, remove h and then append 2*h and 2*h+1;

A080079(n) = T(n,1); T(n,T(n,1)) = 1.

			.   1:   1
.   2:   2   1
.   3:   1   4   3
.   4:   4   3   2   1
.   5:   3   2   1   8   7
.   6:   2   1   8   7   6   5
.   7:   1   8   7   6   5   4   3
.   8:   8   7   6   5   4   3   2   1
.   9:   7   6   5   4   3   2   1  16  15
.  10:   6   5   4   3   2   1  16  15  14  13
.  11:   5   4   3   2   1  16  15  14  13  12  11
.  12:   4   3   2   1  16  15  14  13  12  11  10   9 .

Reinhard Zumkeller, Rows n = 1..128 of triangle, flattened

Cf. A062383 (row maxima).

a240769 n k = a240769_tabl !! (n-1) !! (k-1)
a240769_row n = a240769_tabl !! (n-1)
a240769_tabl = iterate (\(x:xs) -> xs ++ [2*x, 2*x-1]) [1]

A109849 a(1) = 1, a(2) = 2; a(n) = lcm(n, a(n-2)), a(n+1) = lcm((n+1), a(n-3)) and so on until a(2n-1) = lcm(2n-1, a(1)). Then a(2n) = lcm(2n, a(2n-2)) and so on.

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 21, 8, 9, 40, 231, 12, 65, 28, 15, 16, 17, 144, 285, 140, 1365, 132, 5313, 120, 225, 104, 189, 84, 145, 60, 93, 32, 33, 544, 3255, 180, 5365, 1596, 2457, 520, 9225, 840, 228459, 132, 4095, 3220, 13395, 144, 833, 400, 255, 364, 3445, 108, 1155

Offset: 1

Amarnath Murthy, Jul 06 2005

nonn

Another bootstrap sequence. 12 appears twice.

12 60 65 104 120 132 144 252 255 400 912 1008 1020 1980 2112 2457 4092 4095 ... are the smallest numbers that appear twice, with 4095 being the smallest number that appears three times. - Joshua Zucker, May 04 2006

Ivan Neretin, Table of n, a(n) for n = 1..8193

Cf. A109850.

For n > 2, a(n) = lcm(n, a(A080079(n-2))). - Ivan Neretin, Apr 30 2016

More terms from Joshua Zucker, May 04 2006

A353157 a(n) is the distance from n to the nearest integer whose binary expansion has no common 1-bits with that of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25

Offset: 0

Rémy Sigrist, Apr 27 2022

Equivalently the distance to the nearest integer that can be added without carries in base 2.

			For n = 42 ("101010" in binary):
- 21 ("10101") is the greatest number <= 42 that has no common 1-bits with 42,
- 128 ("1000000") is the least number >= 42 that has no common 1-bits with 42,
- so a(42) = min(42-21, 128-42) = min(21, 86) = 21.

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

Cf. A006257, A020988, A097110, A080079, A108019, A353158.

a(n) = { my (high=2^#binary(n), low=high-1-n); min(n-low, high-n) }

a(n) = min(A006257(n), A080079(n)) for any n > 0.
a(n) = 1 iff n belongs to A097110.
a(n) = n/2 iff n belongs to A020988.
a(n) = n/4 iff n belongs to A108019.
2*a(n) - a(2*n) = 0 or 1.

A049929 a(n) = a(1) + a(2) + ... + a(n-1) - a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n-1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 4.

Original entry on oeis.org

1, 3, 4, 5, 12, 20, 41, 83, 168, 254, 550, 1121, 2250, 4507, 9015, 18031, 36064, 54098, 117212, 238932, 480121, 961371, 1923313, 3846922, 7693930, 15387945, 30775932, 61551885, 123103778, 246207563, 492415127, 984830255, 1969660512

Offset: 1

Clark Kimberling

nonn

Empirical: Lim_{n->infinity} a(n+1)/a(n) = 2. - Iain Fox, Dec 05 2017

			For n = 4, 2^p < 3 <= 2^(p+1), so p = 1, m = 2^2 + 2 - 4 = 2, and a(n) = a(1) + a(2) + a(3) - a(2) = 1 + 3 + 4 - 3 = 5.
For n = 6, 2^p < 5 <= 2^(p+1), so p = 2, m = 2^3 + 2 - 6 = 4, and a(n) = a(1) + a(2) + a(3) + a(4) + a(5) - a(4) = 1 + 3 + 4 + 5 + 12 - 5 = 20.

Iain Fox, Table of n, a(n) for n = 1..3325
Iain Fox, Table of n, a(n) for n = 1..8000

Cf. A080079, A049933, A049937, A049945.

Fold[Append[#1, Total@ #1 - #1[[2^Ceiling@ Log2@ #2 + 1 - #2]] ] &, {1, 3, 4}, Range[3, 32]] (* Michael De Vlieger, Dec 06 2017 *)

first(n)= my(res = vector(n), s = 8); res[1]=1; res[2]=3; res[3]=4; for(x=4, n, res[x] = s - res[2*2^logint(x-2, 2)+2-x]; s += res[x]); res; \\ Iain Fox, Dec 05 2017

a(n) = (Sum_{i=1..n-1} a(i)) - a(2^ceiling(log_2(n-1)) + 2 - n) for n > 3. - Iain Fox, Dec 06 2017

For n > 3, a(n) is the sum of all previous terms except a(A080079(n-2)). - Iain Fox, Dec 13 2017

Name edited by Petros Hadjicostas, Nov 06 2019

A122872 Table by antidiagonals, T(n,k) is k-th number that starts with n in binary representation.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 5, 6, 4, 5, 8, 7, 8, 5, 6, 9, 12, 9, 10, 6, 7, 10, 13, 16, 11, 12, 7, 8, 11, 14, 17, 20, 13, 14, 8, 9, 16, 15, 18, 21, 24, 15, 16, 9, 10, 17, 24, 19, 22, 25, 28, 17, 18, 10, 11, 18, 25, 32, 23, 26, 29, 32, 19, 20, 11, 12, 19, 26, 33, 40, 27, 30, 33, 36, 21, 22, 12

Offset: 1

Franklin T. Adams-Watters, Oct 23 2006

In rows n through 2n-1, every integer >= n occurs exactly once.

			Top left corner is:
1 2 3 4 5
2 4 5 8 9
3 6 7 12 13
4 8 9 16 17
5 10 11 20 21

Rows: A000027, A004754, A004755, A004756, A004757, A004758, A004759. Algebraically, A053645 would be row zero, minus A080079 would be row minus one. See also A053644.

T(n,1) = n; T(n,2k) = 2T(n,k); T(n,2k+1) = 2T(n,k) + 1. T(n,k) = k + (n-1) * 2^floor(log_2(k)) = k + (n-1)*A053644(k).

A153587 a(n) = n mod (A062383(n) - n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 4, 1, 0, 1, 0, 0, 0, 2, 4, 6, 8, 10, 2, 5, 0, 4, 2, 2, 0, 2, 0, 0, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 1, 4, 7, 10, 13, 0, 4, 8, 12, 4, 9, 4, 1, 0, 1, 4, 4, 0, 1, 0, 0, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42

Offset: 0

J. Z. Bradley (jzbradley(AT)gmail.com), Dec 29 2008

Michel Marcus, Table of n, a(n) for n = 0..1024

Cf. A062383, A080079.

int A062383(int n) { if(n==0) return 1; return 2*(A062383(n/2)); }
int a(int n) { return n % (A062383(n)-n); }

b(n) = if (n, 2*b(n\2), 1);
a(n) = n % (n - b(n)); \\ Michel Marcus, Jan 28 2018

def A153587(n): return n % ((1 << n.bit_length())-n) # Chai Wah Wu, Jun 30 2022

a(n) = n mod A080079(n), for n > 0. - Michel Marcus, Jan 28 2018

More terms from Michel Marcus, Jan 28 2018

A262617 First differences of A256266.

Original entry on oeis.org

0, 6, 12, 6, 24, 18, 12, 6, 48, 42, 36, 30, 24, 18, 12, 6, 96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 192, 186, 180, 174, 168, 162, 156, 150, 144, 138, 132, 126, 120, 114, 108, 102, 96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 384, 378, 372, 366, 360, 354, 348, 342, 336, 330, 324, 318

Offset: 0

Omar E. Pol, Oct 02 2015

Number of cells turned ON at n-th stage of the cellular automaton of A256266.

			With the terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
0;
6;
12, 6;
24, 18, 12, 6;
48, 42, 36, 30, 24, 18, 12, 6;
96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6;
...
Apart from the initial zero the rows list the initial terms of the positive multiples of 6 in decreasing order.

Cf. A008588, A011782, A080079, A255748, A256257, A256266.

a(n) = if(n==0, 0, -6*n+12*2^floor(log(n)/log(2)));
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

a(n) = 6 * A080079(n), n >= 1.

cp's OEIS Frontend

A327489 T(n, k) = 1 + NOR(k - 1, n - k), where NOR is the Peirce arrow operating bitwise on the inputs, triangle read by rows, T(n, k) for n >= 1, 1 <= k <= n.

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Maple

A097327 Least positive integer m such that m*n has greater decimal digit length than n.

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A101267 a(1) = 1; a(n) = a(2^ceiling(log_2(n)) + 1 - n)th smallest positive integer not yet in the sequence.

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A240769 Triangle read by rows: T(1,1) = 1; T(n+1,k) = T(n,k+1), 1 <= k < n; T(n+1,n) = 2*T(n,1); T(n+1,n+1) = 2*T(n,1) - 1.

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Haskell

A109849 a(1) = 1, a(2) = 2; a(n) = lcm(n, a(n-2)), a(n+1) = lcm((n+1), a(n-3)) and so on until a(2n-1) = lcm(2n-1, a(1)). Then a(2n) = lcm(2n, a(2n-2)) and so on.

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A353157 a(n) is the distance from n to the nearest integer whose binary expansion has no common 1-bits with that of n.

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A049929 a(n) = a(1) + a(2) + ... + a(n-1) - a(m) for n >= 4, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n-1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 4.

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A122872 Table by antidiagonals, T(n,k) is k-th number that starts with n in binary representation.

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A240769 Triangle read by rows: T(1,1) = 1; T(n+1,k) = T(n,k+1), 1 <= k < n; T(n+1,n) = 2T(n,1); T(n+1,n+1) = 2T(n,1) - 1.