A372112 Rad-transform of the squarefree numbers A005117 (see Comments).

1, 2, 3, 5, 1, 7, 1, 11, 13, 1, 1, 17, 19, 1, 1, 23, 1, 29, 1, 31, 1, 1, 1, 37, 1, 1, 41, 1, 43, 1, 47, 1, 53, 1, 1, 1, 59, 61, 1, 1, 1, 67, 1, 1, 71, 73, 1, 1, 1, 79, 1, 83, 1, 1, 1, 89, 1, 1, 1, 1, 97, 101, 1, 103, 1, 1, 107, 109, 1, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 1

Offset

1,2

Comments

For a sequence A with terms a(1), a(2), a(3).... , let R(0) = 1, and for k >= 1 let R(k) = rad(a(1)*a(2)*...*a(k)). Define the Rad-transform of A to be R(n)/R(n-1); n >= 1, where rad is A007947. This sequence is the Rad transform of the squarefree numbers, A = A005117; see Example.

The sequence consists of only 1's and primes.

Sequence is obtained directly from A005117 by leaving all primes and a(1) = 1 untouched, and replacing all composite squarefree numbers with 1. Alternatively: in A000027, delete all squarefull numbers (A013929), replace all squarefree composites with 1, leave primes untouched and concatenate.

Links

Table of n, a(n) for n=1..80.

Example

For A005117, R(k) (k >= 0) is 1 U A128040. That is, 1,1,2,6,30,30,210,... from which: a(1) = 1/1 = 1, a(2) = 2/1 = 2, a(3) = 6/2 = 3, 4(4) = 30/6 = 5, a(5) = 30/30 = 1, and so on. Note that the first term of R(k) is 1, the empty product (product of the first 0 terms of A005117).

Mathematica

k = r = s = 1; f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]]; Reap[Do[While[! SquareFreeQ[k], k++]; s = f[s*f[k]]; Sow[s/r]; r = s; k++, {n, 120}] ][[-1, 1]] (* Michael De Vlieger, Apr 19 2024 *)

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = my(v = select(issquarefree, [1..nn])); my(w = vector(#v, k, rad(prod(i=1, k, v[i])))); concat(1, vector(#w-1, k, w[k+1]/w[k])); \\ Michel Marcus, Apr 19 2024
(Python)
from math import isqrt
from sympy import mobius, isprime
def A372112(n):
    def f(x): return int(n-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m if isprime(m) else 1 # Chai Wah Wu, Dec 23 2024

Crossrefs

Cf. A000040, A005117, A007947, A089026, A128040.

Sequence in context: A139047 A343376 A210945 * A191795 A330612 A121053

Adjacent sequences: A372109 A372110 A372111 * A372113 A372114 A372115

Keyword

nonn

Author

David James Sycamore, Apr 19 2024

Extensions

More terms from Michel Marcus, Apr 19 2024

Status

approved