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

%I A372112 #20 Jul 02 2025 21:17:52
%S A372112 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,
%T A372112 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,
%U A372112 1,1,97,101,1,103,1,1,107,109,1,1,113,1,1,1,1,1,1,127,1,1
%N A372112 Rad-transform of the squarefree numbers A005117 (see Comments).
%C A372112 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.
%C A372112 The sequence consists of only 1's and primes.
%C A372112 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.
%H A372112 Michael De Vlieger, <a href="/A372112/b372112.txt">Table of n, a(n) for n = 1..10000</a>
%e A372112 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).
%t A372112 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 *)
%o A372112 (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o A372112 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
%o A372112 (Python)
%o A372112 from math import isqrt
%o A372112 from sympy import mobius, isprime
%o A372112 def A372112(n):
%o A372112     def f(x): return int(n-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
%o A372112     m, k = n, f(n)
%o A372112     while m != k: m, k = k, f(k)
%o A372112     return m if isprime(m) else 1 # _Chai Wah Wu_, Dec 23 2024
%Y A372112 Cf. A000040, A005117, A007947, A089026, A128040.
%K A372112 nonn
%O A372112 1,2
%A A372112 _David James Sycamore_, Apr 19 2024
%E A372112 More terms from _Michel Marcus_, Apr 19 2024