A346488 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(n) = 0 if mu(n) = -1, and f(n) = n for all other numbers (with mu = Möbius mu, A008683).

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

Offset: 1

Author

Antti Karttunen, Aug 20 2021

Keywords

• nonn

Comments

Restricted growth sequence transform of the sequence f(n) = 0 if mu(n) = -1, and f(n) = n for mu(n) >= 0.

For all i, j:

A305800(i) = A305800(j) => a(i) = a(j) => A305980(i) = A305980(j),

a(i) = a(j) => b(i) = b(j), where b is the pointwise sum of any two multiplicative sequences c and d that are Dirichlet inverses of each other. For example, b can be a sequence like A319340, A323885, or A347094.

Links

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

Crossrefs

Cf. A008683, A070549, A030059 (positions of 2's).

Cf. also A305800, A305980, A319340, A323885, A347094.

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; };
Aux346488(n) = if(moebius(n)<0,0,n);
v346488 = rgs_transform(vector(up_to, n, Aux346488(n)));
A346488(n) = v346488[n];

PARI

A070549(n) = sum(k=1,n,(-1==moebius(k)));
A346488(n) = if(1==n,1,if(-1==moebius(n),2,1+n-A070549(n)));

Formula

a(1) = 1, and for n > 1, if A008683(n) = -1, a(n) = 2, otherwise a(n) = 1 + n - A070549(n).