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

Author

Antti Karttunen, Dec 26 2018

Keywords

• nonn

Comments

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

Links

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

Crossrefs

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).

Programs

PARI

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];

Formula

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