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A286609 Numbers k for which the binary representation of the primes that divide k (A087207) is more than k.

%I A286609 #23 Jul 31 2025 14:31:42
%S A286609 7,11,13,17,19,23,26,29,31,34,37,38,41,43,46,47,51,53,57,58,59,61,62,
%T A286609 67,69,71,73,74,76,79,82,83,86,87,89,92,93,94,95,97,101,103,106,107,
%U A286609 109,111,113,114,115,116,118,122,123,124,127,129,131,133,134,137,138,139,141,142,145,146,148,149,151,155
%N A286609 Numbers k for which the binary representation of the primes that divide k (A087207) is more than k.
%C A286609 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 A286608.
%H A286609 Antti Karttunen, <a href="/A286609/b286609.txt">Table of n, a(n) for n = 1..10000</a>
%H A286609 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A286609 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%t A286609 b[n_] := If[n= 1, 0, Total[2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]];
%t A286609 filterQ[n_] := b[n] >= n;
%t A286609 Select[Range[1000], filterQ] (* _Jean-François Alcover_, Dec 31 2020 *)
%o A286609 (PARI)
%o A286609 A007947(n) = factorback(factorint(n)[, 1]);
%o A286609 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ After _Michel Marcus_
%o A286609 A087207(n) = A048675(A007947(n));
%o A286609 isA286608(n) = (A087207(n) < n);
%o A286609 n=0; j=1; k=1; while(k <= 10000, n=n+1; if(isA286608(n), write("b286608.txt", j, " ", n); j=j+1, write("b286609.txt", k, " ", n); k=k+1));
%o A286609 (Scheme)
%o A286609 ;; With _Antti Karttunen_'s IntSeq-library.
%o A286609 (define A286609 (MATCHING-POS 1 1 (lambda (n) (> (A087207 n) n))))
%o A286609 (Python)
%o A286609 from sympy import factorint, primepi
%o A286609 def a(n):
%o A286609     f=factorint(n)
%o A286609     return sum([2**primepi(i - 1) for i in f])
%o A286609 print([n for n in range(1, 201) if a(n)>n]) # _Indranil Ghosh_, Jun 20 2017
%Y A286609 Cf. A087207, A285315, A285316, A286611.
%Y A286609 Cf. A286608 (complement).
%K A286609 nonn,base
%O A286609 1,1
%A A286609 _Antti Karttunen_, Jun 20 2017