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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.
6
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
1,2
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
FORMULA
a(n) = A323161(n+1) - 1.
PROG
(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];
CROSSREFS
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).
Sequence in context: A319339 A322807 A322805 * A366384 A291761 A319337
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 26 2018
STATUS
approved