A349574 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344696(i) = A344696(j) and A344697(i) = A344697(j), for all i, j >= 1.

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

Offset: 1

Author

Antti Karttunen, Nov 22 2021

Keywords

• nonn

Comments

Restricted growth sequence transform of the ordered pair [A344696(n), A344697(n)].

For all i, j, A003557(i) = A003557(j) => a(i) = a(j); in other words, this sequence is a function of A003557. This follows because A344696(n) = A344696(A057521(n)), A344697(n) = A344696(A057521(n)), and A057521(n) = A064549(A003557(n)).

Apparently, A081770 gives the positions of 2's, which occur on those n where A344696(n) = 7 and A344697(n) = 6.

Links

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

Crossrefs

Cf. A000203, A001615, A003557, A005117 (positions of 1's), A057521, A064549, A081770, A280292, A344695, A344696, A344697.

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; };
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
Aux349574(n) = { my(s=sigma(n),u=A001615(n),g=gcd(u,s)); [s/g, u/g]; };
v349574 = rgs_transform(vector(up_to, n, Aux349574(n)));
A349574(n) = v349574[n];

Formula

For all n >= 1, a(n) = a(A057521(n)). [See comments]