OFFSET
1,1
COMMENTS
m is in this list if and only if v_2(d) + s_2(m) <= m where v_2(d) is the 2-adic valuation of the denominator of sum(i=1..n, 1/(i*2^i)) and s_2(m) is the sum of the digits in the expansion of m in base 2. - Peter Luschny, May 19 2014
LINKS
Robert Israel and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 220 terms from Israel)
A. Straub, V. H. Moll, T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, arXiv:0811.2028 [math.NT], 2008.
Nicolas Wider, Integrality of factorial ratios, Master's Thesis ETH Zürich, 2012.
Wadim Zudilin, Integer-valued factorial ratios, MathOverflow question 26336, 2010.
EXAMPLE
3 is in the sequence because 3!*(1/1/2^1 + 1/2/2^2 + 1/3/2^3) = 4 is an integer. - Robert Israel, May 18 2014
MAPLE
select(k -> type(k!*add(1/i/2^i, i=1..k), integer), [$1..10000]); # Robert Israel, May 18 2014
MATHEMATICA
Select[Range[2000], IntegerQ[#!*Sum[1/(i*2^i), {i, 1, #}]]&] (* Jean-François Alcover, Jul 14 2018 *)
PROG
(Sage)
def is_A069119(n):
s = add(1/(i*2^i) for i in (1..n))
vf = n - sum(ZZ(n).digits(base=2))
return valuation(denominator(s), 2) <= vf
filter(is_A069119, range(1112)) # Peter Luschny, May 19 2014
(PARI) sm(n)=my(s, o); forstep(i=n, 1, -1, o=-valuation(s+=1/(i<<i), 2); if(i+#binary(i)-1<o, return(o))); o
is(n)=hammingweight(n)+sm(n) <= n \\ Charles R Greathouse IV, May 19 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Apr 07 2002
STATUS
approved