A353523 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349905(i) = A349905(j) and A003557(i) = A003557(j), for all i, j >= 1.

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

Offset: 1

Author

Antti Karttunen, Apr 27 2022

Keywords

• nonn

Comments

Restricted growth sequence transform of the ordered pair [A003557(n), A349905(n)], or equally, of the ordered pair [A003415(A003961(n)), A003557(A003961(n))].

This is a prime-shifted variant of A344025, as this is the restricted growth sequence transform of A344025(A003961(n)).

For all i, j:

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

a(i) = a(j) => A349905(i) = A349905(j) => A008836(i) = A008836(j),

a(i) = a(j) => A353571(i) = A353571(j).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003415, A003557, A003961, A008836, A305800, A344025, A349905, A353571.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
Aux353523(n) = { my(s=A003961(n)); [A003415(s), A003557(s)]; };
v353523 = rgs_transform(vector(up_to, n, Aux353523(n)));
A353523(n) = v353523[n];