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A285319 Squarefree numbers n for which A019565(n) < n and A048675(n) is also squarefree.

%I A285319 #23 Jul 18 2025 16:46:59
%S A285319 66,129,130,258,514,1034,1041,1042,2049,2054,2055,2066,2082,2114,4098,
%T A285319 4101,4102,4130,4161,4162,4226,4353,4354,4610,5122,8193,8198,8202,
%U A285319 8205,8206,8210,8211,8229,8259,8706,9218,9219,12291
%N A285319 Squarefree numbers n for which A019565(n) < n and A048675(n) is also squarefree.
%C A285319 Any finite cycle in A019565, if such cycles exist at all, must have at least one member that occurs somewhere in this sequence. Furthermore, such a number n should satisfy A019565(n) < n and that A048675(n)^k is squarefree for all k >= 0.
%t A285319 lim = 4000;
%t A285319 A019565 = Table[Times @@ Prime@Flatten@Position[#, 1] &@
%t A285319    Reverse@IntegerDigits[n, 2], {n, 1, lim}]; (* From Michael De Vlieger in A019565 *)
%t A285319 A048675 = Table[Total[#[[2]]*2^(PrimePi[#[[1]]] - 1) & /@ FactorInteger[n]], {n, 1, lim}]; (* From Jean-François Alcover in A048675 *)
%t A285319 Select[Range[lim], A019565[[#]] < # && SquareFreeQ[#] &&
%t A285319 SquareFreeQ[A048675[[#]]] &] (* _Robert Price_, Apr 07 2019 *)
%o A285319 (PARI)
%o A285319 allocatemem(2^30);
%o A285319 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 A285319 A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ _Michel Marcus_, Oct 10 2016
%o A285319 isA285319(n) = (issquarefree(n) & (A019565(n) < n) && issquarefree(A048675(n)));
%o A285319 n=0; k=0; while(k <= 60, n=n+1; if(isA285319(n),print1(n,", ");k=k+1));
%o A285319 (Scheme)
%o A285319 ;; With _Antti Karttunen_'s IntSeq-library.
%o A285319 (define A285319 (MATCHING-POS 1 0 (lambda (n) (and (< (A019565 n) n) (not (zero? (A008683 n))) (not (zero? (A008683 (A048675 n))))))))
%Y A285319 Subsequence of A285317.
%Y A285319 Cf. A008683, A019565, A048675.
%Y A285319 Cf. also A285320 and discussion in A285331 and A285332.
%K A285319 nonn,hard,more
%O A285319 1,1
%A A285319 _Antti Karttunen_, Apr 18 2017