cp's OEIS Frontend

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

%I A285315 #23 Mar 29 2025 10:53:12
%S A285315 8,16,32,33,64,65,66,128,129,130,131,132,136,256,257,258,259,260,261,
%T A285315 264,272,512,513,514,515,516,517,518,520,521,528,544,576,640,768,1024,
%U A285315 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 A285315 Numbers n for which A019565(n) < n.
%C A285315 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 A285315 Antti Karttunen, <a href="/A285315/b285315.txt">Table of n, a(n) for n = 1..10000</a>
%t A285315 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 A285315 (PARI)
%o A285315 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 A285315 isA285315(n) = (A019565(n) < n);
%o A285315 n=0; k=1; while(k <= 10000, n=n+1; if(isA285315(n),write("b285315.txt", k, " ", n);k=k+1));
%o A285315 (Scheme)
%o A285315 ;; With _Antti Karttunen_'s IntSeq-library.
%o A285315 (define A285315 (MATCHING-POS 1 0 (lambda (n) (< (A019565 n) n))))
%o A285315 (Python)
%o A285315 from sympy import prime, prod
%o A285315 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 A285315 [n for n in range(1, 3001) if a019565(n)<n] # _Indranil Ghosh_, Apr 18 2017, after _Chai Wah Wu_
%Y A285315 Complement: A285316.
%Y A285315 Cf. A019565, A048675.
%Y A285315 Cf. A285317, A285319 (subsequences).
%K A285315 nonn
%O A285315 1,1
%A A285315 _Antti Karttunen_, Apr 18 2017