A285315 - OEIS

%I #19 Jun 10 2020 10:53:28

%S 8,16,32,33,64,65,66,128,129,130,131,132,136,256,257,258,259,260,261,

%T 264,272,512,513,514,515,516,517,518,520,521,528,544,576,640,768,1024,

%U 1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1040,1041,1042,1056,1057,1088,1089,1152,1280,1536,2048,2049,2050,2051

%N Numbers n for which A019565(n) < n.

%C 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 .. ∞.

%H Antti Karttunen, <a href="/A285315/b285315.txt">Table of n, a(n) for n = 1..10000</a>

%t a019565[n_]:=Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2] ; Select[Range[3000], a019565[#]<# &] (* _Indranil Ghosh_, Apr 18 2017, after _Michael De Vlieger_ *)

%o (PARI)

%o 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 isA285315(n) = (A019565(n) < n);

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

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A285315 (MATCHING-POS 1 0 (lambda (n) (< (A019565 n) n))))

%o (Python)

%o from sympy import prime, prod

%o 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

%o [n for n in range(1, 3001) if a019565(n)<n] # _Indranil Ghosh_, Apr 18 2017, after _Chai Wah Wu_

%Y Complement: A285316.

%Y Cf. A019565, A048675.

%Y Cf. A285317, A285319 (subsequences).

%K nonn

%O 1,1

%A _Antti Karttunen_, Apr 18 2017