A069119 Numbers n such that n!*Sum_{i=1..n} 1/(i*2^i) is an integer.

3, 7, 15, 23, 31, 47, 63, 79, 87, 95, 127, 143, 151, 159, 186, 191, 215, 223, 255, 271, 279, 287, 319, 343, 351, 383, 415, 447, 471, 511, 527, 535, 543, 575, 599, 607, 639, 671, 698, 703, 727, 767, 799, 831, 895, 959, 964, 1023, 1039, 1047, 1055, 1087, 1111

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

Cf. A069120.

Sequence in context: A165469 A160160 A192122 * A261413 A187220 A067317

Adjacent sequences: A069116 A069117 A069118 * A069120 A069121 A069122

Keyword

nonn

Author

Benoit Cloitre, Apr 07 2002

Status

approved