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%I A285317 #20 May 07 2021 08:08:54
%S A285317 33,65,66,129,130,131,257,258,259,514,515,517,518,521,1027,1030,1031,
%T A285317 1033,1034,1041,1042,1057,2049,2051,2053,2054,2055,2059,2065,2066,
%U A285317 2081,2082,2113,2114,2177,2305,2561,3073,4097,4098,4099,4101,4102,4103,4105,4106,4109,4115,4129,4130,4161,4162,4226,4353,4354,4609,4610,5122
%N A285317 Squarefree numbers n for which A019565(n) < n.
%C A285317 Any finite cycle in A019565, if such cycles exist at all, must have at least one member that occurs somewhere in this sequence, although certainly not all terms of this sequence could occur in a finite cycle. Specifically, such a number n must occur also in subsequence A285319, and in general, it should satisfy A019565(n) < n and that A048675^{k}(n) is squarefree for all k = 0 .. ∞.
%H A285317 Antti Karttunen, <a href="/A285317/b285317.txt">Table of n, a(n) for n = 1..10000</a>
%F A285317 a(n) = A019565(A285318(n)).
%t A285317 a019565[n_]:=Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2] ; Select[Range[5200], SquareFreeQ[#] && a019565[#]<# &] (* _Indranil Ghosh_, Apr 18 2017, after _Michael De Vlieger_ *)
%o A285317 (PARI)
%o A285317 A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from _M. F. Hasler_
%o A285317 isA285317(n) = (issquarefree(n) & (A019565(n) < n));
%o A285317 n=0; k=1; while(k <= 10000, n=n+1; if(isA285317(n),write("b285317.txt", k, " ", n);k=k+1));
%o A285317 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A285317 (define A285317 (MATCHING-POS 1 0 (lambda (n) (and (< (A019565 n) n) (not (zero? (A008683 n)))))))
%o A285317 (Python)
%o A285317 from operator import mul
%o A285317 from functools import reduce
%o A285317 from sympy import prime
%o A285317 from sympy.ntheory.factor_ import core
%o A285317 def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1
%o A285317 print([n for n in range(1, 5201) if core(n) == n and a019565(n) < n]) # _Indranil Ghosh_, Apr 18 2017, after _Chai Wah Wu_
%Y A285317 Intersection of A005117 and A285315.
%Y A285317 Cf. A005117, A008683, A019565, A048675, A285318, A285319 (subsequence), A285332.
%K A285317 nonn
%O A285317 1,1
%A A285317 _Antti Karttunen_, Apr 18 2017

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