A364225 a(n) = number of k <= m such that rad(k) | m, where m = A025487(n) and rad(n) = A007947(n).

1, 2, 3, 5, 4, 8, 5, 11, 18, 6, 14, 15, 26, 7, 18, 20, 36, 8, 23, 44, 25, 68, 26, 49, 9, 29, 58, 31, 96, 32, 65, 10, 35, 76, 38, 131, 39, 83, 84, 11, 88, 42, 156, 43, 97, 45, 174, 46, 104, 106, 12, 111, 50, 283, 206, 51, 121, 53, 228, 54, 130, 133, 13, 138, 58

Offset

1,2

Comments

Not a permutation of natural numbers: a(4) = a(7) = 5.

Let S_rad(m) be the sequence { k : rad(k) | rad(m) }. This sequence gives the number of k <= rad(m). Seen another way, this sequence gives the position of m in S_rad(m).

The number m appears after its factors in S_rad(m). If k < sqrt(m) then k^2 also appears before m.

Scatterplot exhibits trajectories according to t = omega(m) = A001221(A025487(n)). The first term in each trajectory is A002110(t).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1809 from Michael De Vlieger)

Michael De Vlieger, Log log scatterplot of a(n), n = 1..1809, accentuating primorial A025487(n) in red, noting a(n) above in black, and labeling n in blue italic below for the first 24 terms.

Formula

a(n) = A010846(A025487(n)).

Example

a(1) = 1 since 1 is the only number k that does not exceed 1 such that rad(k) | 1.
a(2) = 2 since k in {1, 2} are such that rad(k) | 2.
a(3) = 3 since k in {1, 2, 4} are such that rad(k) | 4.
a(4) = 5 since k in {1, 2, 3, 4, 6} are such that rad(k) | 6, etc.

Mathematica

rad[x_] := Times @@ FactorInteger[#][[All, 1]];
Map[Function[{n, r},
Count[Range[n], _?(Divisible[r, rad[#]] &)]] @@ {#, rad[#]} &,
  {1}~Join~Select[Range[Times @@ Prime@ Range[6]],
     # == Transpose@ {Prime@ Range[Length[#]], ReverseSort[#[[All, -1]] ]} &@
     FactorInteger[#] &] ]

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 A364225(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 1 if x == 1 else sum(g2(x//ps[m]**i, m-1) for i in range(integer_log(x, ps[m])[0]+1)) if m else integer_log(x, ps[0])[0]+1
    return g2(k, len(ps)-1) # Chai Wah Wu, Apr 16 2026

Crossrefs

Cf. A001221, A002110, A007947, A010846, A025487, A363061.

Sequence in context: A347423 A255234 A394753 * A256995 A249129 A065561

Adjacent sequences: A364222 A364223 A364224 * A364226 A364227 A364228

Keyword

nonn

Author

Michael De Vlieger, Oct 24 2023

Status

approved