A336471 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A336158(i) = A336158(j), for all i, j >= 1.

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 3, 3, 5, 1, 2, 4, 6, 2, 7, 3, 6, 2, 4, 3, 8, 3, 6, 5, 6, 1, 7, 2, 7, 4, 6, 6, 7, 2, 3, 7, 9, 3, 10, 6, 9, 2, 11, 4, 5, 3, 6, 8, 7, 3, 12, 6, 9, 5, 6, 6, 13, 1, 7, 7, 9, 2, 12, 7, 9, 4, 6, 6, 10, 6, 12, 7, 9, 2, 14, 3, 6, 7, 5, 9, 12, 3, 6, 10, 12, 6, 12, 9, 12, 2, 3, 11, 13, 4, 6, 5, 6, 3, 15

Offset: 1

Author

Antti Karttunen, Jul 22 2020

Keywords

• nonn

Comments

Restricted growth sequence transform of the ordered pair [A329697(n), A336158(n)].

For all i, j:

A336470(i) = A336470(j) => a(i) = a(j)

a(i) = a(j) => A336396(i) = A336396(j),

a(i) = a(j) => A336469(i) = A336469(j) => A336477(i) = A336477(j).

This sequence has an ability to see where the terms of A003401 are, as they are the indices of zeros in A336469. Specifically, they are numbers k that satisfy the condition A329697(k) = A001221(A336158(k)), i.e., numbers for which A329697(k) is equal to the number of distinct prime divisors of the odd part of k. See also comments in array A334100.

Links

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

Crossrefs

Cf. A000265, A003401, A046523, A329697, A334100, A336158.

Cf. also A003602, A324400, A336159, A336160, A336396, A336469, A336470, A336472, A336473, A336477.

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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
Aux336471(n) = [A329697(n), A336158(n)];
v336471 = rgs_transform(vector(up_to, n, Aux336471(n)));
A336471(n) = v336471[n];