A323234 -id:A323234 - cp's OEIS frontend

A323235 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, f(n) = -1 if n is an odd number > 1, and f(n) = A323234(n) for even numbers >= 4.

1, 2, 3, 2, 3, 4, 3, 2, 3, 5, 3, 6, 3, 7, 3, 2, 3, 5, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 2, 3, 5, 3, 8, 3, 9, 3, 14, 3, 15, 3, 16, 3, 17, 3, 18, 3, 19, 3, 20, 3, 21, 3, 22, 3, 23, 3, 24, 3, 25, 3, 2, 3, 5, 3, 8, 3, 9, 3, 14, 3, 15, 3, 16, 3, 17, 3, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 32, 3, 33, 3, 34, 3, 35, 3, 36, 3, 37, 3, 38, 3

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

For all i, j:
  A319701(i) = A319701(j) => a(i) = a(j),
  A323234(i) = A323234(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A053645, A079944, A319701, A323234.

Cf. also A323241 (somewhat analogous filter sequence for prime factorization).

up_to = 65537;
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; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A079944off0(n) = (1==binary(2+n)[2]);
A323235aux(n) = if(1==n,0,if(n%2,-1,[A053645(n), A079944off0(n-2)]));
v323235 = rgs_transform(vector(up_to,n,A323235aux(n)));
A323235(n) = v323235[n];

A323236 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, f(n) = -1 if n is an even number > 2, and f(n) = A323234(n) for odd numbers >= 3.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 4, 2, 6, 2, 7, 2, 8, 2, 4, 2, 6, 2, 9, 2, 10, 2, 11, 2, 12, 2, 13, 2, 14, 2, 4, 2, 6, 2, 9, 2, 10, 2, 15, 2, 16, 2, 17, 2, 18, 2, 19, 2, 20, 2, 21, 2, 22, 2, 23, 2, 24, 2, 25, 2, 26, 2, 4, 2, 6, 2, 9, 2, 10, 2, 15, 2, 16, 2, 17, 2, 18, 2, 27, 2, 28, 2, 29, 2, 30, 2, 31, 2, 32, 2, 33, 2, 34, 2, 35, 2, 36, 2, 37, 2, 38, 2, 39

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

For all i, j:
  A319702(i) = A319702(j) => a(i) = a(j),
  A323234(i) = A323234(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A053645, A079944, A319702, A323234.

Cf. also A323242 (somewhat analogous filter sequence for prime factorization).

up_to = 65537;
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; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A079944off0(n) = (1==binary(2+n)[2]);
A323236aux(n) = if(1==n,0,if(!(n%2),-1,[A053645(n), A079944off0(n-2)]));
v323236 = rgs_transform(vector(up_to,n,A323236aux(n)));
A323236(n) = v323236[n];

A324400 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n = 2^k and k > 0, and f(n) = n for all other numbers.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 2, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Antti Karttunen, Mar 01 2019

nonn

In the following, A stands for this sequence, A324400, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j >= 1: S(i) = S(i) => T(i) = T(j).
For example, the following chains of implications hold:
  A -> A286619 -> A005811,
and
  A -> A003602 -> A286622 -> A000120,
               -> A286378 -> A002487,
               -> A323889,
               -> A000593,
               -> A001227,
among many others.

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

Cf. A000523.
Cf. A000120, A000265, A000593, A001227, A002487, A003602, A005811, A209229, A286378, A286619, A286622, A294898, A297159, A318311, A323234, A323889, A323904, A324343, A324345 (some of the matching sequences).
Cf. also A103391, A295300, A305800, A305801, A324401.

A000523(n) = if(n<1, 0, #binary(n)-1);
A324400(n) = if(n<4,n,if(!bitand(n,n-1),2,1+n-A000523(n)));

If n <= 3, a(n) = n; and for n >= 4, if A209229(n) = 1, then a(n) = 2, otherwise a(n) = 1 + n - A000523(n).

cp's OEIS Frontend

A323235 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, f(n) = -1 if n is an odd number > 1, and f(n) = A323234(n) for even numbers >= 4.

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A323236 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, f(n) = -1 if n is an even number > 2, and f(n) = A323234(n) for odd numbers >= 3.

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A324400 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n = 2^k and k > 0, and f(n) = n for all other numbers.

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