A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 8, 2, 12, 7, 13, 4, 14, 8, 11, 1, 15, 9, 15, 5, 16, 10, 17, 3, 18, 11, 19, 6, 20, 8, 15, 2, 21, 12, 22, 7, 23, 13, 22, 4, 24, 14, 25, 8, 26, 11, 27, 1, 28, 15, 29, 9, 30, 15, 22, 5, 31, 16, 32, 10, 30, 17, 24, 3, 33, 18, 28, 11, 34, 19, 35, 6, 36, 20, 37, 8, 38, 15, 35, 2, 39, 21, 40, 12, 41, 22, 42, 7, 43

Offset: 1

Author

Antti Karttunen, Feb 11 2022

Keywords

• nonn

Comments

Restricted growth sequence transform of the ordered pair [A206787(n), A336651(n)], or equally, of sequence b(n) = A291750(A000265(n)).

For all i, j >= 1:

A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),

A324400(i) = A324400(j) => A351460(i) = A351460(j) => a(i) = a(j),

a(i) = a(j) => A000593(i) = A000593(j),

a(i) = a(j) => A347385(i) = A347385(j),

a(i) = a(j) => A351037(i) = A351037(j) => A347240(i) = A347240(j).

From Antti Karttunen, Nov 23 2023: (Start)

Conjectured to be equal to the lexicographically earliest infinite sequence such that b(i) = b(j) => A000593(i) = A000593(j) and A336467(i) = A336467(j) for all i, j >= 1. In any case, a(i) = a(j) => b(i) = b(j) for all i, j >= 1 [because both A000593(n) and A336467(n) can be computed from the values of A206787(n) and A336651(n)], but whether the implication holds to the opposite direction is still open. Empirically this has been checked up to n = 2^22. See also comment in A351040.

(End)

Links

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

Crossrefs

Cf. A206787, A336651.

Cf. also A000593, A003602, A291750, A291751, A324400, A336467, A347240, A347385, A351040, A351460.

Differs from A351037 for the first time at n=103, where a(103) = 42 while A351037(103) = 27.

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; };
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ From A206787
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351461(n) = [A206787(n), A336651(n)];
v351461 = rgs_transform(vector(up_to, n, Aux351461(n)));
A351461(n) = v351461[n];