A289813 -id:A289813 - cp's OEIS frontend

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

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

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];

A366793 Binary encoding of the ones in the balanced ternary representation of Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 3, 1, 0, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 4, 0, 2, 0, 1, 2, 0, 0, 3, 0, 1, 0, 2, 1, 0, 1, 2, 2, 0, 0, 3, 0, 2, 0, 1, 0, 3, 2, 0, 3, 0, 3, 4, 1, 0, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 4, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 3, 1, 2, 1, 0, 2, 1, 0, 4, 1, 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 positive coefficients, and converting them to a binary string gives "10", which in base-2 (A007088) is equal to 2, therefore a(254) = 2.

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, A289813, A323909, A366794.

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)); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A366793(n) = A289813(A323909(n));

a(n) = A289813(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).

A356248 Image of 1 under repeated application of the map k -> (2k-1,2k,2k-1).

Original entry on oeis.org

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

Offset: 0

Arie Bos, Jul 31 2022

nonn

			1 --> 1 2 1 --> 1 2 1 3 4 3 1 2 1 --> 1 2 1 3 4 3 1 2 1 5 6 5 7 8 7 5 6 5 1 2 1 3 4 3 1 2 1 -->...

Cf. A289813.

a(n) = fromdigits(digits(n,3)%2,2) + 1; \\ Kevin Ryde, Jul 31 2022

def aupton(terms):
    a, n = [1], 0
    while len(a) < 3*terms: a, n = a + [(1<Michael S. Branicky, Jul 31 2022

If A(n) = (a(0),a(1),...,a(3^n-1)), then A(n+1) = (A(n),2^n+A(n),A(n)).

a(n) = A289813(n) + 1. - Rémy Sigrist, Jul 31 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A366793 Binary encoding of the ones in the balanced ternary representation of Per Nørgård's "infinity sequence".

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A356248 Image of 1 under repeated application of the map k -> (2k-1,2k,2k-1).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Python

Formula