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

Author

Antti Karttunen, Jun 20 2017

Keywords

• nonn
• base

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

Crossrefs

Cf. A087207, A285315, A285316.

Cf. A286609 (complement).

Intersection with A286611 gives A286612.

Programs

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 *)

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));

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)Indranil Ghosh, Jun 20 2017

Scheme

;; With Antti Karttunen's IntSeq-library.
(define A286608 (MATCHING-POS 1 1 (lambda (n) (< (A087207 n) n))))