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A286608 Numbers k for which the binary representation of the primes that divide k (A087207) is less than k.

%I A286608 #28 Jul 31 2025 14:33:24
%S A286608 1,2,3,4,5,6,8,9,10,12,14,15,16,18,20,21,22,24,25,27,28,30,32,33,35,
%T A286608 36,39,40,42,44,45,48,49,50,52,54,55,56,60,63,64,65,66,68,70,72,75,77,
%U A286608 78,80,81,84,85,88,90,91,96,98,99,100,102,104,105,108,110,112,117,119,120,121,125,126,128,130
%N A286608 Numbers k for which the binary representation of the primes that divide k (A087207) is less than k.
%C A286608 Any finite cycle of A087207, if such cycles exist at all, should have at least one term that is a member of this sequence, and also at least one term that is a member of A286609.
%H A286608 Antti Karttunen, <a href="/A286608/b286608.txt">Table of n, a(n) for n = 1..10000</a>
%H A286608 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A286608 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%t A286608 b[n_] := If[n==1, 0, Total[2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]];
%t A286608 filterQ[n_] := b[n] < n;
%t A286608 Select[Range[1000], filterQ] (* _Jean-François Alcover_, Dec 31 2020 *)
%o A286608 (PARI)
%o A286608 A007947(n) = factorback(factorint(n)[, 1]);
%o A286608 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ After _Michel Marcus_
%o A286608 A087207(n) = A048675(A007947(n));
%o A286608 isA286608(n) = (A087207(n) < n);
%o A286608 n=0; j=1; k=1; while(j <= 10000, n=n+1; if(isA286608(n), write("b286608.txt", j, " ", n); j=j+1, write("b286609.txt", k, " ", n); k=k+1));
%o A286608 (Scheme)
%o A286608 ;; With _Antti Karttunen_'s IntSeq-library.
%o A286608 (define A286608 (MATCHING-POS 1 1 (lambda (n) (< (A087207 n) n))))
%o A286608 (Python)
%o A286608 from sympy import factorint, primepi
%o A286608 def a(n):
%o A286608     f=factorint(n)
%o A286608     return sum([2**primepi(i - 1) for i in f])
%o A286608 print([n for n in range(1, 201) if a(n)<n]) # _Indranil Ghosh_, Jun 20 2017
%Y A286608 Cf. A087207, A285315, A285316.
%Y A286608 Cf. A286609 (complement).
%Y A286608 Intersection with A286611 gives A286612.
%K A286608 nonn,base
%O A286608 1,2
%A A286608 _Antti Karttunen_, Jun 20 2017