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

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

Offset

1,2

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences related to binary expansion of n

Index entries for sequences computed from indices in prime factorization

Mathematica

b[n_] := If[n==1, 0, Total[2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]];
filterQ[n_] := b[n] < n;
Select[Range[1000], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

Prog

(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ After Michel Marcus
A087207(n) = A048675(A007947(n));
isA286608(n) = (A087207(n) < n);
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));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A286608 (MATCHING-POS 1 1 (lambda (n) (< (A087207 n) n))))
(Python)
from sympy import factorint, primepi
def a(n):
    f=factorint(n)
    return sum([2**primepi(i - 1) for i in f])
print([n for n in range(1, 201) if a(n)<n]) # Indranil Ghosh, Jun 20 2017

Crossrefs

Cf. A087207, A285315, A285316.

Cf. A286609 (complement).

Intersection with A286611 gives A286612.

Sequence in context: A229461 A230709 A080671 * A225736 A225735 A225734

Adjacent sequences: A286605 A286606 A286607 * A286609 A286610 A286611

Keyword

nonn,base

Author

Antti Karttunen, Jun 20 2017

Status

approved