A393167 Number of m <= s(n) with the same squarefree kernel as s(n), where s = A025487.

1, 1, 2, 1, 3, 2, 4, 4, 1, 5, 5, 6, 2, 6, 8, 9, 4, 7, 11, 6, 13, 1, 14, 7, 8, 15, 10, 17, 2, 18, 12, 9, 20, 15, 22, 4, 23, 18, 19, 10, 20, 25, 6, 26, 23, 28, 8, 29, 26, 27, 11, 28, 31, 1, 11, 32, 33, 34, 14, 35, 36, 38, 12, 40, 38, 2, 18, 39, 44, 45, 41, 22, 47

Offset

1,3

Comments

Length of row A025487(n) of A369609.

Scatterplot shows trajectories i associated with rad(s(n)) = A002110(i), beginning at n = A098719(i).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14.

Formula

a(n) = A008479(A025487(n)).

a(A098719(n)) = 1.

a(A025488(n)) = n.

a(n) < A364225(n) for n > 1; a(1) = A364225(1) = 1.

Example

Table of n, s(n), and a(n) for n = 1..12, listing numbers m < s(n) such that rad(m) = rad(s(n)), where rad = A007947:
   n  s(n) a(n)  row s(n) of A369609.
  --------------------------------------
   1    1    1   {1}
   2    2    1   {2}
   3    4    2   {2, 4}
   4    6    1   {6}
   5    8    3   {2, 4, 8}
   6   12    2   {6, 12}
   7   16    4   {2, 4, 8, 16}
   8   24    4   {6, 12, 18, 24}
   9   30    1   {30}
  10   32    5   {2, 4, 8, 16, 32}
  11   36    5   {6, 12, 18, 24, 36}
  12   48    6   {6, 12, 18, 24, 36, 48}

Mathematica

fQ[x_] := Or[x == 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, ReverseSort[FactorInteger[x][[All, -1]] ] ] == x]; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Map[(r = rad[#]; Count[Range[#], _?(rad[#] == r &)]) &, Select[Range[2^13], fQ] ]
(* Alternative: load function f from A025487 and theta from the Mathematica link in A369609, then: *)
Map[theta[-#] &, Union@ Flatten@ f[6]]

Prog

(Python)
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log, primefactors
from oeis_sequences.OEISsequences import bisection
def A393167(n):
    if n == 1: return 1
    @lru_cache(maxsize=None)
    def g1(x, m, j): return sum(g1(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, p = n-1+x, 1
        for k in count(1):
            p *= prime(k)
            if p>x:
                break
            c -= g1(x, k, 1)
        return c
    ps = tuple(sorted(primefactors(k:=bisection(f, n, n))))
    @lru_cache(maxsize=None)
    def g2(x, m): return 0 if x == 1 else sum(g2(x//ps[m]**i, m-1) for i in range(1, integer_log(x, ps[m])[0]+1)) if m else integer_log(x, ps[0])[0]
    return g2(k, len(ps)-1) # Chai Wah Wu, Apr 16 2026

Crossrefs

Cf. A002110, A007947, A008479, A025487, A025488 (index of powers of primes in A025487), A098719 (index of squarefree terms in A025487), A364225 (number of m <= s(n) with rad(m) | s(n)), A369609.

Sequence in context: A034390 A391228 A368671 * A183912 A144693 A328399

Adjacent sequences: A393164 A393165 A393166 * A393168 A393169 A393170

Keyword

nonn,look

Author

Michael De Vlieger, Apr 12 2026

Status

approved