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%I #23 Mar 28 2022 11:41:32
%S 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,
%T 15,14,16,15,16,17,17,18,18,19,3,19,20,4,20,21,21,22,5,22,23,23,24,25,
%U 26,24,27,28,29,30,25,26,6,27,7,31,28,29,8,32,30,9
%N The n-th squarefree number is the a(n)-th squarefree number having its number of primes.
%C 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.
%H David A. Corneth, <a href="/A340313/b340313.txt">Table of n, a(n) for n = 1..10000</a>
%H Alois P. Heinz, <a href="/A340313/a340313.jpg">Plot of n, a(n) for n = 1..1000000</a>
%F a(n) = #{x|x <= n, A072047(x) = A072047(n)}.
%e {x|x <= 6, A072047(x) = A072047(6) = 1} = {2,3,4,6}, therefore a(6) = 4.
%e {x|x <= 28, A072047(x) = A072047(28) = 3} = {19,28}, therefore a(28) = 2.
%p with(numtheory):
%p b:= proc(n) option remember; local k; if n=1 then 1 else
%p for k from 1+b(n-1) while not issqrfree(k) do od; k fi
%p end:
%p p:= proc() 0 end:
%p a:= proc(n) option remember; local h; a(n-1);
%p h:= bigomega(b(n)); p(h):= p(h)+1;
%p end: a(0):=0:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jan 06 2021
%t b[n_] := b[n] = Module[{k}, If[n == 1, 1,
%t For[k = 1 + b[n - 1], !SquareFreeQ[k], k++]; k]];
%t p[_] = 0;
%t a[n_] := a[n] = Module[{h}, a[n - 1];
%t h = PrimeOmega[b[n]]; p[h] = p[h]+1];
%t a[0] = 0;
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 28 2022, after _Alois P. Heinz_ *)
%o (Haskell)
%o a340313 n = a340313_list !! (n-1)
%o a340313_list = repetitions a072047_list
%o where
%o repetitions [] = []
%o repetitions (a:as) = 1 : h a as (repetitions as)
%o h _ [] _ = []
%o h b (c:cs) (r:rs) = (if c == b then succ else id) r : h b cs rs
%o (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
%Y Cf. A001221, A001222, A005117 (squarefree numbers), A058933, A067003, A072047 (number of prime factors), A340316 (squarefree numbers array).
%K nonn
%O 1,3
%A _Peter Dolland_, Jan 04 2021
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