A323235 -id:A323235 - cp's OEIS frontend

A324401 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n is an odd prime, f(n) = -2 if n = 2^k, with k > 1, and f(n) = n for all other numbers.

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

Offset: 1

Antti Karttunen, Mar 01 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A305976(i) = A305976(j) => A001221(i) = A001221(j),
  a(i) = a(j) => A322591(i) = A322591(j),
  a(i) = a(j) => A323235(i) = A323235(j),
  a(i) = a(j) => A324399(i) = A324399(j),
  a(i) = a(j) => A297159(i) = A297159(j).

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

Cf. A000523, A000720.

Cf. also A297159, A305801, A305976, A322591, A323235, A324399, A324400.

A000523(n) = if(n<1, 0, #binary(n)-1);
A324401(n) = if(n<4,n,if(isprime(n),3,if(!bitand(n,n-1),4,4+n-A000523(n)-primepi(n))));

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; };
Aux324401(n) = if((n>2) && (isprime(n)||!bitand(n,n-1)),-(2-(n%2)),n);
\\ Equally: Aux324401(n) = if(n<=2,n,if(isprime(n),-1,if(!bitand(n,n-1),-2,n)));
v324401 = rgs_transform(vector(up_to, n, Aux324401(n)));
A324401(n) = v324401[n];

If n <= 2, a(n) = n, for n > 2, if n is an odd prime, a(n) = 3, if n = 2^k, with k >= 2, a(n) = 4, otherwise a(n) = 4 + n - A000523(n) - A000720(n).

A323241 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=2) = -n, f(n) = 0 if n is an odd number > 1, and f(n) = A300226(n) for even numbers >= 4.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 5, 3, 7, 3, 5, 3, 8, 3, 9, 3, 7, 3, 5, 3, 10, 3, 5, 3, 7, 3, 11, 3, 12, 3, 5, 3, 13, 3, 5, 3, 10, 3, 11, 3, 7, 3, 5, 3, 14, 3, 15, 3, 7, 3, 16, 3, 10, 3, 5, 3, 17, 3, 5, 3, 18, 3, 11, 3, 7, 3, 19, 3, 20, 3, 5, 3, 7, 3, 11, 3, 14, 3, 5, 3, 17, 3, 5, 3, 10, 3, 21, 3, 7, 3, 5, 3, 22, 3, 23, 3, 24, 3, 11, 3, 10, 3

Offset: 1

Antti Karttunen, Jan 07 2019

nonn

For all i, j:
  A319701(i) = A319701(j) => a(i) = a(j),
  a(i) = a(j) => A007814(i) = A007814(j).
  a(i) = a(j) => A183063(i) = A183063(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A007814, A052126, A183063, A300226, A319701, A319988, A323235, A323237, A323238, A323242.

up_to = 10000;
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; };
A052126(n) = if(1==n, n, n/vecmax(factor(n)[, 1]));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A323241aux(n) = if(n<=2,-n,if(n%2,0,[A052126(n), A319988(n)]));
v323241 = rgs_transform(vector(up_to,n,A323241aux(n)));
A323241(n) = v323241[n];

A323234 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, and for n > 1, f(n) = ordered pair [A053645(n), A079944(n-2)], where A053645(n) gives n without its most significant bit, while A079944(n-2) gives the second most significant bit of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 08 2019

Also the restricted growth sequence transform of function f(1) = 0, f(n) = [A053645(n), A278222(n)] for n > 1.
For all i, j:
  a(i) = a(j) => A286622(i) = A286622(j),
  a(i) = a(j) => A323235(i) = A323235(j),
  a(i) = a(j) => A323236(i) = A323236(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, A278222, A286622, A323235, A323236.

Cf. also A300226 (an 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]);
A323234aux(n) = if(1==n,0,[A053645(n), A079944off0(n-2)]);
v323234 = rgs_transform(vector(up_to,n,A323234aux(n)));
A323234(n) = v323234[n];

cp's OEIS Frontend

A324401 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n is an odd prime, f(n) = -2 if n = 2^k, with k > 1, and f(n) = n for all other numbers.

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A323241 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=2) = -n, f(n) = 0 if n is an odd number > 1, and f(n) = A300226(n) for even numbers >= 4.

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A323234 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, and for n > 1, f(n) = ordered pair [A053645(n), A079944(n-2)], where A053645(n) gives n without its most significant bit, while A079944(n-2) gives the second most significant bit of n.

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