A351040 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336158(i) = A336158(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, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 17, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 25, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 17, 32, 9, 33, 17, 34, 5, 35, 18, 36, 10, 33, 19, 37, 3, 38, 20, 39, 11, 40, 21, 41, 6, 42

Offset: 1

Author

Antti Karttunen, Jan 31 2022

Keywords

• nonn
• easy

Comments

Restricted growth sequence transform of the ordered triplet [A336158(n), A206787(n), A336651(n)].

For all i, j >= 1:

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

a(i) = a(j) => A336390(i) = A336390(j) => A336391(i) = A336391(j),

a(i) = a(j) => A347374(i) = A347374(j),

a(i) = a(j) => A351036(i) = A351036(j) => A113415(i) = A113415(j),

a(i) = a(j) => A351461(i) = A351461(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), A336158(i) = A336158(j) and A336467(i) = A336467(j), for all i, j >= 1 (this was the original definition). In any case it holds that a(i) = a(j) => b(i) = b(j) for all i, j >= 1. See comment in A351461.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65539

Crossrefs

Cf. A000265, A000593, A336158, A336467.

Cf. also A003602, A336390, A336391, A351037, A366891.

Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.

Differs from A351035 for the first time at n=175, where a(175) = 80, while A351035(175) = 78.

Differs from A351036 for the first time at n=637, where a(637) = 272, while A351036(637) = 261.

Programs

PARI

up_to = 65539;
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]); };
A336158(n) = A046523(A000265(n));
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d));
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351040(n) = [A336158(n), A206787(n), A336651(n)];
v351040 = rgs_transform(vector(up_to, n, Aux351040(n)));
A351040(n) = v351040[n];

Extensions

Original definition moved to the comment section and replaced with a definition that is at least as encompassing, and conjectured to be equal to the original one. - Antti Karttunen, Nov 23 2023