A322822 -id:A322822 - cp's OEIS frontend

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

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

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

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; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

A322988 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0 if n is a prime power > 2, f(2) = -1, and f(n) = A322990(n) for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 02 2019

nonn

For all i, j: a(i) = a(j) => A322989(i) = A322989(j).

For all i, j > 2: A305976(i) = A305976(j) => a(i) = a(j).

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

Cf. A289271, A289272, A305976, A322989, A322990.

Cf. A322805, A322822 for analogous constructions for filter sequences.

up_to = 8192;
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; };
A289271(n) = { my(v=0,i=0,x=1); for(d=2,oo,if(n==1, return(v)); if(1==gcd(x,d)&&1==omega(d), if(!(n%d)&&1==gcd(d,n/d), v += 2^i; n /= d; x *= d); i++)); }; \\ After Rémy Sigrist's program for A289271.
A289272(n) = { my(m=1, pp=1); while(n>0, pp++; while(!isprimepower(pp)||(gcd(pp,m)>1), pp++); if(n%2, m *= pp); n >>=1); (m); }; \\ Antti Karttunen, Jan 02 2019
A322990(n) = A289272(A289271(n)>>1);
A322988aux(n) = if(2==n,-1,if(isprimepower(n),0,A322990(n)));
v322988 = rgs_transform(vector(up_to,n,A322988aux(n)));
A322988(n) = v322988[n];

A323082 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -(n mod 2) if n is a prime, and f(n) = A300840(n) for any other number.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

For all i, j: A323074(i) = A323074(j) => a(i) = a(j).
Like the related A322822 also this filter sequence satisfies the following two implications, for all i, j >= 1:
  a(i) = a(j) => A322356(i) = A322356(j),
  a(i) = a(j) => A290105(i) = A290105(j).

Antti Karttunen, Table of n, a(n) for n = 1..65539

Cf. A052330, A300840, A305801, A322822, A323074.

Cf. also A290105, A322356.

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; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A050376list(up_to) = { my(v=vector(up_to), i=0); for(n=1,oo,if(A302777(n), i++; v[i] = n); if(i == up_to,return(v))); };
v050376 = A050376list(up_to);
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&A302777(n/d), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A300840(n) = A052330(A052331(n)>>1);
A323082aux(n) = if(isprime(n),-(n%2),A300840(n));
v323082 = rgs_transform(vector(up_to,n,A323082aux(n)));
A323082(n) = v323082[n];

A322823 a(n) = 0 if n is 1 or a Fermi-Dirac prime (A050376), otherwise a(n) = 1 + a(A300840(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 1, 0, 3, 2, 1, 0, 2, 0, 1, 2, 3, 0, 2, 0, 1, 2, 1, 4, 3, 0, 1, 2, 3, 0, 2, 0, 3, 4, 1, 0, 2, 0, 1, 2, 3, 0, 2, 4, 3, 2, 1, 0, 3, 0, 1, 5, 3, 4, 2, 0, 3, 2, 4, 0, 3, 0, 1, 2, 3, 5, 2, 0, 4, 0, 1, 0, 3, 4, 1, 2, 3, 0, 4, 5, 3, 2, 1, 4, 2, 0, 1, 6, 3, 0, 2, 0, 3, 4

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

For n > 1, a(n) gives the number of edges needed to traverse from n to reach the leftmost branch (where the terms of A050376 are located) in the binary tree illustrated in A052330.

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

Cf. A050376, A052330, A052331, A302777, A300840, A322822.

up_to = 10000;
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A050376list(up_to) = { my(v=vector(up_to), i=0); for(n=1,oo,if(A302777(n), i++; v[i] = n); if(i == up_to,return(v))); };
v050376 = A050376list(up_to);
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A300840(n) = A052330(A052331(n)>>1);
A322823(n) = if((1==n)||(1==A302777(n)),0,1+A322823(A300840(n)));

a(1) = 0; for n > 1, if A302777(n) == 1, a(n) = 0, otherwise a(n) = 1 + a(A300840(n)).

cp's OEIS Frontend

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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A322988 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0 if n is a prime power > 2, f(2) = -1, and f(n) = A322990(n) for all other numbers.

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A323082 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -(n mod 2) if n is a prime, and f(n) = A300840(n) for any other number.

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A322823 a(n) = 0 if n is 1 or a Fermi-Dirac prime (A050376), otherwise a(n) = 1 + a(A300840(n)).

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