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A340313
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The n-th squarefree number is the a(n)-th squarefree number having its number of primes.
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3
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1, 1, 2, 3, 1, 4, 2, 5, 6, 3, 4, 7, 8, 5, 6, 9, 7, 10, 1, 11, 8, 9, 10, 12, 11, 12, 13, 2, 14, 13, 15, 14, 16, 15, 16, 17, 17, 18, 18, 19, 3, 19, 20, 4, 20, 21, 21, 22, 5, 22, 23, 23, 24, 25, 26, 24, 27, 28, 29, 30, 25, 26, 6, 27, 7, 31, 28, 29, 8, 32, 30, 9
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OFFSET
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1,3
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COMMENTS
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The sequence gives the column index of A005117(n) in the array A340316 and may be understood as a complementary addition to A072047 giving the row index.
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LINKS
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David A. Corneth, Table of n, a(n) for n = 1..10000
Alois P. Heinz, Plot of n, a(n) for n = 1..1000000
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FORMULA
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a(n) = #{x|x <= n, A072047(x) = A072047(n)}.
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EXAMPLE
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{x|x <= 6, A072047(x) = A072047(6) = 1} = {2,3,4,6}, therefore a(6) = 4.
{x|x <= 28, A072047(x) = A072047(28) = 3} = {19,28}, therefore a(28) = 2.
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MAPLE
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with(numtheory):
b:= proc(n) option remember; local k; if n=1 then 1 else
for k from 1+b(n-1) while not issqrfree(k) do od; k fi
end:
p:= proc() 0 end:
a:= proc(n) option remember; local h; a(n-1);
h:= bigomega(b(n)); p(h):= p(h)+1;
end: a(0):=0:
seq(a(n), n=1..100); # Alois P. Heinz, Jan 06 2021
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MATHEMATICA
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b[n_] := b[n] = Module[{k}, If[n == 1, 1,
For[k = 1 + b[n - 1], !SquareFreeQ[k], k++]; k]];
p[_] = 0;
a[n_] := a[n] = Module[{h}, a[n - 1];
h = PrimeOmega[b[n]]; p[h] = p[h]+1];
a[0] = 0;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 28 2022, after Alois P. Heinz *)
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PROG
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(Haskell)
a340313 n = a340313_list !! (n-1)
a340313_list = repetitions a072047_list
where
repetitions [] = []
repetitions (a:as) = 1 : h a as (repetitions as)
h _ [] _ = []
h b (c:cs) (r:rs) = (if c == b then succ else id) r : h b cs rs
(PARI) first(n) = {v = vector(5); n--; res = vector(n); t = 0; for(i = 2, oo, f = factor(i)[, 2]; if(vecmax(f) == 1, if(#f > #v, v = concat(v, vector(#f - #v)) ); t++; v[#f]++; res[t] = v[#f]; if(t >= n, return(concat(1, res)) ) ) ) } \\ David A. Corneth, Jan 07 2021
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CROSSREFS
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Cf. A001221, A001222, A005117 (squarefree numbers), A058933, A067003, A072047 (number of prime factors), A340316 (squarefree numbers array).
Sequence in context: A152201 A265579 A336879 * A120873 A125161 A331791
Adjacent sequences: A340310 A340311 A340312 * A340314 A340315 A340316
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KEYWORD
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nonn
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AUTHOR
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Peter Dolland, Jan 04 2021
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STATUS
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approved
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