A285315 Numbers n for which A019565(n) < n.

8, 16, 32, 33, 64, 65, 66, 128, 129, 130, 131, 132, 136, 256, 257, 258, 259, 260, 261, 264, 272, 512, 513, 514, 515, 516, 517, 518, 520, 521, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1033, 1034, 1040, 1041, 1042, 1056, 1057, 1088, 1089, 1152, 1280, 1536, 2048, 2049, 2050, 2051

Offset: 1

Author

Antti Karttunen, Apr 18 2017

Keywords

• nonn

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 consecutively nested subsequences A285317, 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

Crossrefs

Complement: A285316.

Cf. A019565, A048675.

Cf. A285317, A285319 (subsequences).

Programs

Mathematica

a019565[n_]:=Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2] ; Select[Range[3000], a019565[#]<# &] (* Indranil Ghosh, Apr 18 2017, after Michael De Vlieger *)

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
isA285315(n) = (A019565(n) < n);
n=0; k=1; while(k <= 10000, n=n+1; if(isA285315(n),write("b285315.txt", k, " ", n);k=k+1));

Python

from sympy import prime, prod
def a019565(n): return prod(prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1') if n > 0 else 1
[n for n in range(1, 3001) if a019565(n)Indranil Ghosh, Apr 18 2017, after Chai Wah Wu

Scheme

;; With Antti Karttunen's IntSeq-library.
(define A285315 (MATCHING-POS 1 0 (lambda (n) (< (A019565 n) n))))