A285317 Squarefree numbers n for which A019565(n) < n.

33, 65, 66, 129, 130, 131, 257, 258, 259, 514, 515, 517, 518, 521, 1027, 1030, 1031, 1033, 1034, 1041, 1042, 1057, 2049, 2051, 2053, 2054, 2055, 2059, 2065, 2066, 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

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, 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 .. ∞.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = A019565(A285318(n)).

Mathematica

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 *)

Prog

(PARI)
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
isA285317(n) = (issquarefree(n) & (A019565(n) < n));
n=0; k=1; while(k <= 10000, n=n+1; if(isA285317(n), write("b285317.txt", k, " ", n); k=k+1));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A285317 (MATCHING-POS 1 0 (lambda (n) (and (< (A019565 n) n) (not (zero? (A008683 n)))))))
(Python)
from operator import mul
from functools import reduce
from sympy import prime
from sympy.ntheory.factor_ import core
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
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

Crossrefs

Intersection of A005117 and A285315.

Cf. A005117, A008683, A019565, A048675, A285318, A285319 (subsequence), A285332.

Sequence in context: A116350 A331071 A062059 * A198592 A325377 A061560

Adjacent sequences: A285314 A285315 A285316 * A285318 A285319 A285320

Keyword

nonn

Author

Antti Karttunen, Apr 18 2017

Status

approved