A325273 Prime omicron of n!.

0, 0, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

0,4

Comments

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

The prime omicron of n (A304465) is 0 if n is 1, 1 if n is prime, and otherwise the second-to-last part of the omega-sequence of n. For example, the prime omicron of 180 is 2.

Conjecture: all terms after a(10) = 4 are less than 4.

From James Rayman, Apr 17 2021: (Start)

The conjecture is false. a(3804) = 4. In fact, there are 91 values of n < 10000 such that a(n) = 4.

The first value of n such that a(n) = 5 is 37934. For any other n < 5*10^5, a(n) < 5. (End)

Links

James Rayman, Table of n, a(n) for n = 0..10000

Mathematica

omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];
omicron[n_]:=Switch[n, 1, 0, _?PrimeQ, 1, _, omseq[n][[-2]]];
Table[omicron[n!], {n, 0, 100}]

Prog

(Python)
from sympy.ntheory import *
def red(v):
    r = {}
    for i in v: r[i] = r.get(i, 0) + 1
    return r
def omicron(v):
    if len(v) == 0: return 0
    if len(v) == 1: return v[0]
    else: return omicron(list(red(v).values()))
f, a_list = {}, []
for i in range(101):
    a_list.append(omicron(list(f.values())))
    g = factorint(i+1)
    for k in g: f[k] = f.get(k, 0) + g[k]
print(a_list) # James Rayman, Apr 17 2021

Crossrefs

a(n) = A055396(A325275(n)/2).

Cf. A000142, A006939, A303555, A323023, A325238, A325272, A325274, A325275, A325276, A325277.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

Sequence in context: A331853 A187184 A301375 * A352541 A359438 A279408

Adjacent sequences: A325270 A325271 A325272 * A325274 A325275 A325276

Keyword

nonn

Author

Gus Wiseman, Apr 18 2019

Extensions

More terms from James Rayman, Apr 17 2021

Status

approved