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A372112
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Rad-transform of the squarefree numbers A005117 (see Comments).
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0
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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