A363455 The number of distinct primorial numbers (A002110) larger than 1 in the representation of A025487(n) as a product of primorial numbers.

0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 3, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 3, 2, 1, 3, 2, 3, 2, 2, 2, 2, 2, 1, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 3, 2, 2, 3, 2, 3, 3

Offset

1,6

Comments

The number of distinct exponents in the prime factorization of A025487(n).

The minimal number of powers of primorial numbers (A100778) in the representation of A025487(n) as a product of powers of primorial numbers.

The record values are all the nonnegative integers. The positions of the records are the positions of the terms of the Chernoff sequence (A006939) in A025487, i.e., the first position of k, for k = 0, 1, 2, ..., is A363456(k).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A071625(A025487(n)).

Mathematica

e[1] = 0; e[n_] := Length[Union[FactorInteger[n][[;; , 2]]]]; s = {0}; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;; , 2]] && PrimePi[f[[-1, 1]]] == Length[f], AppendTo[s, e[n]]], {n, 2, 10000}]; s

Prog

(Python)
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log, primorial, factorint
from oeis_sequences.OEISsequences import bisection
def A363455(n):
    @lru_cache(maxsize=None)
    def g(x, m, j): return sum(g(x//(prime(m)**i), m-1, i) for i in range(j, integer_log(x, prime(m))[0]+1)) if m-1 else max(0, x.bit_length()-j)
    def f(x):
        c = n-1+x
        for k in count(1):
            if primorial(k)>x:
                break
            c -= g(x, k, 1)
        return c
    return len(set(factorint(bisection(f, n, n)).values())) # Chai Wah Wu, Mar 23 2026

Crossrefs

Cf. A002110, A006939, A025487, A051282, A071625, A100778, A304886, A363456.

Sequence in context: A329344 A081652 A302790 * A333849 A140816 A029433

Adjacent sequences: A363452 A363453 A363454 * A363456 A363457 A363458

Keyword

nonn

Author

Amiram Eldar, Jun 03 2023

Status

approved