A289814 -id:A289814 - cp's OEIS frontend

A340383 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(A304759(n)), A278222(A291759(n))], for all i, j >= 1.

1, 2, 1, 3, 4, 5, 2, 6, 7, 3, 3, 8, 1, 9, 2, 10, 11, 6, 4, 12, 11, 13, 3, 14, 9, 15, 3, 16, 12, 17, 3, 18, 3, 3, 7, 19, 3, 9, 19, 19, 6, 3, 5, 8, 12, 20, 1, 21, 8, 22, 12, 23, 11, 24, 12, 25, 6, 8, 26, 27, 12, 13, 12, 28, 13, 29, 4, 12, 9, 20, 26, 30, 31, 22, 10, 16, 5, 14, 6, 32, 33, 8, 3, 12, 10, 23, 15, 14, 19, 8

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

Restricted growth sequence transform of the ordered pair [A340381(n), A340382(n)], or equally, of the function f(n) = A290093(A048673(n)).

For all i, j: a(i) = a(j) => A286586(i) = A286586(j) => A286585(i) = A286585(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A048673, A278222, A290093, A291759, A304759, A340381, A340382, A340684.

Cf. also A290094, A305303.

up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux340383(n) = [A278222(A291759(n)),A278222(A304759(n))];
v340383 = rgs_transform(vector(up_to,n,Aux340383(n)));
A340383(n) = v340383[n];

A374362 a(n) is the least term t of A005836 such that n - t also belongs to A005836.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 3, 3, 4, 0, 0, 1, 0, 0, 1, 3, 3, 4, 9, 9, 10, 9, 9, 10, 12, 12, 13, 0, 0, 1, 0, 0, 1, 3, 3, 4, 0, 0, 1, 0, 0, 1, 3, 3, 4, 9, 9, 10, 9, 9, 10, 12, 12, 13, 27, 27, 28, 27, 27, 28, 30, 30, 31, 27, 27, 28, 27, 27, 28, 30, 30, 31, 36, 36, 37, 36

Offset: 0

Rémy Sigrist, Jul 06 2024

To compute a(n): in the ternary expansion of n, replace 1's by 0's and 2's by 1's.

			The first terms, in decimal and in ternary, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     1       2          1
   3     0      10          0
   4     0      11          0
   5     1      12          1
   6     3      20         10
   7     3      21         10
   8     4      22         11
   9     0     100          0
  10     0     101          0
  11     1     102          1
  12     0     110          0
  13     0     111          0
  14     1     112          1
  15     3     120         10

Rémy Sigrist, Table of n, a(n) for n = 0..6561

Cf. A005836, A289814, A374361.

a(n) = fromdigits(apply(d -> [0, 0, 1][1+d], digits(n, 3)), 3)

from gmpy2 import digits
def A374362(n): return int(digits(n,3).replace('1','0').replace('2','1'),3) # Chai Wah Wu, Jul 09 2024

a(n) = A374361(n, 0).
a(n) = n - A374363(n).
a(n) >= 0 with equality iff n belongs to A374361.
a(n) = A005836(1 + A289814(n)).

A289869 Square array T(n,k) (n>=0, k>=0) read by antidiagonals downwards: T(n,k) = A005836(n) + 2*A005836(k).

Original entry on oeis.org

0, 2, 1, 6, 3, 3, 8, 7, 5, 4, 18, 9, 9, 6, 9, 20, 19, 11, 10, 11, 10, 24, 21, 21, 12, 15, 12, 12, 26, 25, 23, 22, 17, 16, 14, 13, 54, 27, 27, 24, 27, 18, 18, 15, 27, 56, 55, 29, 28, 29, 28, 20, 19, 29, 28, 60, 57, 57, 30, 33, 30, 30, 21, 33, 30, 30, 62, 61, 59

Offset: 1

Rémy Sigrist, Jul 14 2017

If n and k have no common one bit in base 2 representation (n AND k = 0), then n = A289813(T(n,k)) and k = A289814(T(n,k)).

This sequence, when restricted to the pairs of numbers without common bits in base 2 representation, is the inverse of the function n -> (A289813(n), A289814(n)).

			The table begins:
x\y:    0   1   2   3   4   5   6   7   8   9  ...
0:      0   2   6   8   18  20  24  26  54  56 ...
1:      1   3   7   9   19  21  25  27  55  57 ...
2:      3   5   9   11  21  23  27  29  57  59 ...
3:      4   6   10  12  22  24  28  30  58  60 ...
4:      9   11  15  17  27  29  33  35  63  65 ...
5:      10  12  16  18  28  30  34  36  64  66 ...
6:      12  14  18  20  30  32  36  38  66  68 ...
7:      13  15  19  21  31  33  37  39  67  69 ...
8:      27  29  33  35  45  47  51  53  81  83 ...
9:      28  30  34  36  46  48  52  54  82  84 ...
...

Rémy Sigrist, First 100 antidiagonals of array, flattened

Cf. A005836, A289813, A289814.

T(n,k) = fromdigits(binary(n),3) + 2*fromdigits(binary(k),3)

def T(n, k): return int(bin(n)[2:], 3) + 2*int(bin(k)[2:], 3)
for n in range(11): print([T(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 03 2017

A317805 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, a(n) AND a(n + k) <> a(n + 2*k) (where AND denotes the bitwise AND operator).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Aug 07 2018

This sequence has similarities with A276204: here we consider the bitwise AND operator, there the addition operator.
Apparently, the variant where we use the bitwise OR operator corresponds, up to a change of offset, to A289814.
The scatterplot of the sequence has fractal features (see illustrations in Links section).

			For n = 10:
- a(10-2*1) AND a(10-1) = 2 AND 2 = 2,
- a(10-2*2) AND a(10-2) = 1 AND 2 = 0,
- a(10-2*3) AND a(10-3) = 1 AND 1 = 1,
- a(10-2*4) AND a(10-4) = 0 AND 1 = 0,
- hence a(10) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A317805
Rémy Sigrist, Scatterplot of the first 9000000 terms
Rémy Sigrist, Colored scatterplot of the first 9000000 terms (where the color is function of the greatest p such that floor(a(n)/2^p) == 1 mod 4 and n + b(a(n)) >= 2 * b(ceil(n/2^p)*2^p) and b(k) is the least m such that a(m) = k)

Cf. A276204, A289814.

A332413 a(n) is the imaginary part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332412 gives real parts.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 12 2020

			For n = 103:
- 103 = 4*5^2 + 3*5^0,
- so f(123) = 3^2 * i^(4-1) + 3^0 * i^(3-1) = -1 - 9*i,
- and a(n) = -9.

Rémy Sigrist, Table of n, a(n) for n = 0..15624

Cf. A031219, A289814, A332412 (real parts and additional comments).

a(n) = { my (d=Vecrev(digits(n,5))); imag(sum (k=1, #d, if (d[k], 3^(k-1)*I^(d[k]-1), 0))) }

a(n) = 0 iff the n-th row of A031219 has neither 2's nor 4's.
a(5*n)   = 3*a(n).
a(5*n+1) = 3*a(n).
a(5*n+2) = 3*a(n) + 1.
a(5*n+3) = 3*a(n).
a(5*n+4) = 3*a(n) - 1.

A340684 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A291759(n), A278222(A304759(n))], for all i, j >= 1.

Original entry on oeis.org

1, 2, 1, 3, 4, 5, 6, 7, 8, 3, 9, 10, 1, 11, 12, 13, 14, 7, 4, 15, 14, 16, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 9, 3, 8, 26, 24, 11, 27, 26, 28, 3, 29, 30, 22, 31, 1, 32, 33, 34, 35, 36, 14, 37, 35, 38, 39, 10, 40, 41, 22, 42, 43, 44, 45, 46, 4, 15, 47, 31, 40, 48, 49, 50, 51, 21, 52, 53, 54, 55, 56, 30, 24, 15

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

Restricted growth sequence transform of the ordered pair [A291759(n), A278222(A304759(n))].

For all i, j: a(i) = a(j) => A340383(i) = A340383(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A048673, A278222, A290093, A291759, A304759, A340383.

up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux340684(n) = [A291759(n),A278222(A304759(n))];
v340684 = rgs_transform(vector(up_to,n,Aux340684(n)));
A340684(n) = v340684[n];

A366794 Binary encoding of the twos (-1's) in the balanced ternary representation of Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 2, 3, 3, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 2, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 2, 4, 2, 0, 0, 2, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 2, 3, 3, 0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 1, 2, 0, 2, 0, 0, 0, 0, 2, 3, 3, 0, 0

Offset: 0

Antti Karttunen, Oct 24 2023

nonn

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

			A004718(254) = -7. In balanced ternary representation (see A117966) this is represented as -1*9 + 1*3 + -1*1. Taking the negative coefficients, and converting them to a binary string gives "101", which in base-2 (A007088) is equal to 5, therefore a(254) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A004718, A007088, A117966, A289814, A323909, A366793.

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ From the code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A323909(n) = { my(x = A004718(n)); if(x >= 0,A117967(x),A117968(-x)); };
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A366794(n) = A289814(A323909(n));

a(n) = A289814(A323909(n)).

A293450 Restricted growth sequence transform of (3*A293225(n) + A010872(n)), a filter combining (n mod 3) with two products, the other formed from the 1-digits (A293221) and the other from the 2-digits (A293222) present in the ternary expansions of proper divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 6, 7, 8, 9, 2, 10, 6, 11, 12, 13, 2, 14, 6, 15, 16, 17, 2, 18, 19, 20, 21, 22, 2, 23, 6, 24, 25, 26, 27, 28, 6, 29, 30, 31, 2, 32, 6, 33, 34, 35, 2, 36, 37, 38, 14, 39, 2, 40, 41, 42, 43, 44, 2, 45, 6, 46, 47, 48, 49, 50, 6, 51, 52, 53, 2, 54, 6, 55, 56, 57, 58, 59, 6, 60, 61, 62, 2, 63, 64, 65, 66, 67, 2

Offset: 1

Antti Karttunen, Nov 06 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..59049

Cf. A002324, A010872, A293221, A293222, A293225, A293226.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A289814(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };
Anot_submitted(n) = (1/2)*(2 + ((A293222(n) + A293221(n))^2) - A293222(n) - 3*A293221(n)); \\ Eq.class-wise equal to A293225.
Anot2submitted(n) = ((3*Anot_submitted(n))+(n%3));
write_to_bfile(1,rgs_transform(vector(59049,n,Anot2submitted(n))),"b293450.txt");

For all i, j: a(i) = a(j) => A002324(i) = A002324(j).

cp's OEIS Frontend

A340383 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(A304759(n)), A278222(A291759(n))], for all i, j >= 1.

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A374362 a(n) is the least term t of A005836 such that n - t also belongs to A005836.

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A289869 Square array T(n,k) (n>=0, k>=0) read by antidiagonals downwards: T(n,k) = A005836(n) + 2*A005836(k).

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A317805 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, a(n) AND a(n + k) <> a(n + 2*k) (where AND denotes the bitwise AND operator).

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A332413 a(n) is the imaginary part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332412 gives real parts.

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A340684 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A291759(n), A278222(A304759(n))], for all i, j >= 1.

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A366794 Binary encoding of the twos (-1's) in the balanced ternary representation of Per Nørgård's "infinity sequence".

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A293450 Restricted growth sequence transform of (3*A293225(n) + A010872(n)), a filter combining (n mod 3) with two products, the other formed from the 1-digits (A293221) and the other from the 2-digits (A293222) present in the ternary expansions of proper divisors of n.

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