A372575 Lexicographically earliest infinite sequence such that a(i) = a(j) => A276085(i) = A276085(j), for all i, j >= 1, where A276085 is the primorial base log-function.

1, 2, 3, 3, 4, 5, 6, 5, 7, 8, 9, 7, 10, 11, 12, 7, 13, 14, 15, 12, 16, 17, 18, 14, 19, 20, 4, 16, 21, 22, 23, 14, 24, 25, 26, 4, 27, 28, 29, 22, 30, 31, 32, 24, 33, 34, 35, 4, 36, 37, 38, 29, 39, 8, 40, 31, 41, 42, 43, 33, 44, 45, 46, 4, 47, 48, 49, 38, 50, 51, 52, 8, 53, 54, 55, 41, 56, 57, 58, 33, 12, 59, 60, 46, 61, 62, 63, 48, 64, 65, 66, 50, 67, 68, 69, 8

Offset: 1

Author

Antti Karttunen, May 25 2024

Keywords

• nonn

Comments

Restricted growth sequence transform of A276085, where A276085 is fully additive with a(p) = A002110(A000720(p)-1).

For all i, j >= 1:

a(i) = a(j) => A035263(i) = A035263(j),

a(i) = a(j) => A369001(i) = A369001(j),

a(i) = a(j) => A372576(i) = A372576(j).

A048103(n) gives the position of the initial occurrence of each n, while A100716 gives the indices of the further occurrences.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to prime indices in the factorization of n
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A000720, A002110, A276085, A035263, A369001, A372576.

Cf. A048103 (positions of records and initial occurrences), A100716.

Cf. also A322815.

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; };
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
v372575 = rgs_transform(vector(up_to, n, A276085(n)));
A372575(n) = v372575[n];

Formula

For all n >= 1, a(A048103(n)) = n.