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%I #33 Sep 08 2022 08:45:06
%S 1,2,9,10,39,348,1272,10682,18275,414912,5606336,8712340,20920564,
%T 47201552,140142814,240574848,5459371212,16993264107,22955387784,
%U 23807694876,33482496720
%N 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).
%C a(11) > 500000. - _Emeric Deutsch_, Aug 04 2005
%C a(22) > 7*10^11. - _Giovanni Resta_, Nov 29 2019
%e 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.
%p 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
%t Do[d = Divisors[n]; If[IntegerQ[Dot[Range[Length[d]], Map[(1/#)&, d]]], Print[n]], {n, 1, 10^8}] (* _Ryan Propper_, Jul 30 2005 *)
%o (PARI) /* to have b(n)=sum k/d(k) */ b(n)=sum(i=1,numdiv(n),i/component(divisors(n),i))
%o (PARI) isok(n) = my(d=divisors(n)); denominator(sum(k=1, #d, k/d[k])) == 1; \\ _Michel Marcus_, Sep 10 2017
%o (Magma) [k:k in [1..500000]|IsIntegral( &+[m/Divisors(k)[m]:m in [1..#Divisors(k)]])]; // _Marius A. Burtea_, Dec 06 2019
%Y Cf. A027750, A073083.
%K nonn,more
%O 1,2
%A _Benoit Cloitre_, Aug 17 2002
%E 5 more terms from _Ryan Propper_, Jul 30 2005
%E Two further terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 31 2005
%E a(17)-a(21) from _Giovanni Resta_, Nov 29 2019
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