A073082 Numbers n such that sum k/d(k) is an integer, where d(k) is the k-th divisor of n (the divisors of n are in increasing order).

1, 2, 9, 10, 39, 348, 1272, 10682, 18275, 414912, 5606336, 8712340, 20920564, 47201552, 140142814, 240574848, 5459371212, 16993264107, 22955387784, 23807694876, 33482496720

Offset

1,2

Comments

a(11) > 500000. - Emeric Deutsch, Aug 04 2005

a(22) > 7*10^11. - Giovanni Resta, Nov 29 2019

Links

Table of n, a(n) for n=1..21.

Example

Divisors of 39 are [1, 3, 13, 39] and 1/1 + 2/3 + 3/13 + 4/39 = 2 is an integer hence 39 is in the sequence.

Maple

with(numtheory): a:=proc(n) local div: div:=divisors(n): if type(sum(k/div[k], k=1..tau(n)), integer)=true then n else fi end: seq(a(n), n=1..50000); # Emeric Deutsch, Aug 04 2005

Mathematica

Do[d = Divisors[n]; If[IntegerQ[Dot[Range[Length[d]], Map[(1/#)&, d]]], Print[n]], {n, 1, 10^8}] (* Ryan Propper, Jul 30 2005 *)

Prog

(PARI) /* to have b(n)=sum k/d(k) */ b(n)=sum(i=1, numdiv(n), i/component(divisors(n), i))
(PARI) isok(n) = my(d=divisors(n)); denominator(sum(k=1, #d, k/d[k])) == 1; \\ Michel Marcus, Sep 10 2017
(Magma) [k:k in [1..500000]|IsIntegral( &+[m/Divisors(k)[m]:m in [1..#Divisors(k)]])]; // Marius A. Burtea, Dec 06 2019

Crossrefs

Cf. A027750, A073083.

Sequence in context: A058551 A119183 A179888 * A300129 A191401 A363223

Adjacent sequences: A073079 A073080 A073081 * A073083 A073084 A073085

Keyword

nonn,more

Author

Benoit Cloitre, Aug 17 2002

Extensions

5 more terms from Ryan Propper, Jul 30 2005

Two further terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 31 2005

a(17)-a(21) from Giovanni Resta, Nov 29 2019

Status

approved