A286608 - OEIS

%I #25 May 07 2021 00:46:39

%S 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 36,39,40,42,44,45,48,49,50,52,54,55,56,60,63,64,65,66,68,70,72,75,77,

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

%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 A286609.

%H Antti Karttunen, <a href="/A286608/b286608.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(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 (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A286608 (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.

%Y Cf. A286609 (complement).

%Y Intersection with A286611 gives A286612.

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Jun 20 2017