A379240 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j, where f(n) = [A003415(n), A085731(n)] if A359550(n) = 1, otherwise f(n) = n.

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 37, 64

Offset: 1

Author

Antti Karttunen, Dec 19 2024

Keywords

• nonn

Comments

It is conjectured that this is also the lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A376418(i) = A376418(j), for all i, j >= 1, i.e., the restricted growth sequence transform of the triple [A003415(n), A085731(n), A376418(n)]. This is true if for every pair of i and j for which i <> j, and A376418(i) = A376418(j) > 0, the ordered pairs [A003415(i), A085731(i)] and [A003415(j), A085731(j)] differ from each other.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A003415, A048103, A083345, A085371, A100716, A359550, A369051, A376418.

Differs from A344025 first at n=140, where a(140) = 97, while A344025(140) = 92.

Differs from A369046 first at n=171, where a(171) = 63, while A369046(171) = 121.

Programs

PARI

up_to = 100000;
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]));
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
Aux379240(n) = if(!A359550(n), n, my(d=A003415(n)); [d, gcd(d,n)]);
v379240 = rgs_transform(vector(up_to, n, Aux379240(n)));
A379240(n) = v379240[n];

Formula

For all i, j >= 1:

a(i) = a(j) => A369051(i) = A369051(j) => A083345(i) = A083345(j),

a(i) = a(j) => A376418(i) = A376418(j).