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Numbers n for which the binary representation of the primes that divide n (A087207) is more than n.
5

%I #20 May 06 2021 22:11:42

%S 7,11,13,17,19,23,26,29,31,34,37,38,41,43,46,47,51,53,57,58,59,61,62,

%T 67,69,71,73,74,76,79,82,83,86,87,89,92,93,94,95,97,101,103,106,107,

%U 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 Numbers n for which the binary representation of the primes that divide n (A087207) is more than n.

%C 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 Antti Karttunen, <a href="/A286609/b286609.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%t b[n_] := If[n= 1, 0, Total[2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]];

%t filterQ[n_] := b[n] >= n;

%t Select[Range[1000], filterQ] (* _Jean-François Alcover_, Dec 31 2020 *)

%o (PARI)

%o A007947(n) = factorback(factorint(n)[, 1]);

%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ After _Michel Marcus_

%o A087207(n) = A048675(A007947(n));

%o isA286608(n) = (A087207(n) < n);

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

%o (define A286609 (MATCHING-POS 1 1 (lambda (n) (> (A087207 n) n))))

%o (Python)

%o from sympy import factorint, primepi

%o def a(n):

%o f=factorint(n)

%o return sum([2**primepi(i - 1) for i in f])

%o print([n for n in range(1, 201) if a(n)>n]) # _Indranil Ghosh_, Jun 20 2017

%Y Cf. A087207, A285315, A285316, A286611.

%Y Cf. A286608 (complement).

%K nonn,base

%O 1,1

%A _Antti Karttunen_, Jun 20 2017