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

66, 129, 130, 258, 514, 1034, 1041, 1042, 2049, 2054, 2055, 2066, 2082, 2114, 4098, 4101, 4102, 4130, 4161, 4162, 4226, 4353, 4354, 4610, 5122, 8193, 8198, 8202, 8205, 8206, 8210, 8211, 8229, 8259, 8706, 9218, 9219, 12291

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..38.

Mathematica

lim = 4000;
A019565 = Table[Times @@ Prime@Flatten@Position[#, 1] &@
   Reverse@IntegerDigits[n, 2], {n, 1, lim}]; (* From Michael De Vlieger in A019565 *)
A048675 = Table[Total[#[[2]]*2^(PrimePi[#[[1]]] - 1) & /@ FactorInteger[n]], {n, 1, lim}]; (* From Jean-François Alcover in A048675 *)
Select[Range[lim], A019565[[#]] < # && SquareFreeQ[#] &&
SquareFreeQ[A048675[[#]]] &] (* Robert Price, Apr 07 2019 *)

Prog

(PARI)
allocatemem(2^30);
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
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
isA285319(n) = (issquarefree(n) & (A019565(n) < n) && issquarefree(A048675(n)));
n=0; k=0; while(k <= 60, n=n+1; if(isA285319(n), print1(n, ", "); k=k+1));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A285319 (MATCHING-POS 1 0 (lambda (n) (and (< (A019565 n) n) (not (zero? (A008683 n))) (not (zero? (A008683 (A048675 n))))))))

Crossrefs

Subsequence of A285317.

Cf. A008683, A019565, A048675.

Cf. also A285320 and discussion in A285331 and A285332.

Sequence in context: A350204 A031184 A039443 * A044189 A044570 A118163

Adjacent sequences: A285316 A285317 A285318 * A285320 A285321 A285322

Keyword

nonn,hard,more

Author

Antti Karttunen, Apr 18 2017

Status

approved