A322809 - OEIS

%I #45 Feb 09 2019 23:27:42

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%C For all i, j:

%C   A305801(i) = A305801(j) => a(i) = a(j),

%C   a(i) = a(j) => A039636(i) = A039636(j).

%C For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).

%C 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

%H Antti Karttunen, <a href="/A322809/b322809.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

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

%o (PARI)

%o up_to = 65537;

%o 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; };

%o A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));

%o v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));

%o A322809(n) = v322809[n];

%Y Cf. A004526, A039636, A305801, A323161.

%Y 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:

%Y   A322809 (A004526, unless an odd prime) [This sequence],

%Y   A322589 (A007913, unless an odd prime),

%Y   A322591 (A007947, unless an odd prime),

%Y   A322805 (A252463, unless an odd prime),

%Y   A323082 (A300840, unless an odd prime),

%Y   A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),

%Y   A322988 (A322990, unless a prime power > 2),

%Y   A323078 (A097246, unless an odd prime),

%Y   A322808 (A097246, unless a squarefree number > 2),

%Y   A322816 (A048675, unless an odd prime),

%Y   A322807 (A285330, unless an odd prime),

%Y   A322814 (A286621, unless an odd prime),

%Y   A322824 (A242424, unless an odd prime),

%Y   A322973 (A006370, unless an odd prime),

%Y   A322974 (A049820, unless n > 1 and n is in A046642),

%Y   A323079 (A060681, unless an odd prime),

%Y   A322587 (A295887, unless an odd prime),

%Y   A322588 (A291751, unless an odd prime),

%Y   A322592 (A289625, unless an odd prime),

%Y   A323369 (A323368, unless an odd prime),

%Y   A323371 (A295886, unless an odd prime),

%Y   A323374 (A323373, unless an odd prime),

%Y   A323401 (A323372, unless an odd prime),

%Y   A323405 (A323404, unless an odd prime).

%K nonn

%O 1,2

%A _Antti Karttunen_, Dec 26 2018