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

1,1

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 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

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 *)

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
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));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A285315 (MATCHING-POS 1 0 (lambda (n) (< (A019565 n) n))))
(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)<n] # Indranil Ghosh, Apr 18 2017, after Chai Wah Wu

Crossrefs

Complement: A285316.

Cf. A019565, A048675.

Cf. A285317, A285319 (subsequences).

Sequence in context: A018922 A290287 A303981 * A020948 A219547 A259751

Adjacent sequences: A285312 A285313 A285314 * A285316 A285317 A285318

Keyword

nonn

Author

Antti Karttunen, Apr 18 2017

Status

approved