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Least multiple of n whose sum of divisors is divisible by n.
3

%I #29 May 02 2016 17:39:46

%S 1,6,6,12,40,6,28,56,90,40,473,24,117,28,120,336,1139,90,703,120,420,

%T 946,3151,120,3725,234,918,28,5017,120,496,672,891,2176,2660,792,2701,

%U 1406,234,120,6683,420,11051,1892,270,6302,13207,528,2548,3800,3417,2340

%N Least multiple of n whose sum of divisors is divisible by n.

%C See A227470(n) for the sequence a(n)/n. If n = prime(i) is a prime then A097018 gives the answer: a(n) = n*A097018(i). One can show that a(n) always exists - see A227470 for the proof. - _N. J. A. Sloane_, May 01 2016

%H R. J. Mathar, <a href="/A272349/b272349.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = n*A227470(n). - _R. J. Mathar_, May 02 2016

%e For n = 2, a(2) = 6 because it is the smallest number divisible by 2 whose sum of divisors (12) is also divisible by 2; 3 and 5 are not divisible by 2 and the sum of divisors of 2 and 4 is 3 and 7, hence also not divisible by 2.

%p A272349 := proc(n)

%p local k;

%p for k from 1 do

%p if modp(numtheory[sigma](k*n),n) =0 then

%p return k*n;

%p end if;

%p end do:

%p end proc: # _R. J. Mathar_, May 02 2016

%t A272349 = {}; Do[k = n; While[!(Divisible[k, n] && Divisible[DivisorSigma[1, k], n]), k++]; AppendTo[A272349, k], {n, 65}]; A272349

%o (PARI) for(n=1, 65, k=n; while(!(k%n==0&&sigma(k)%n==0), k++); print1(k ", "))

%o (PARI) a(n)=my(k=n); while(sigma(k)%n,k+=n); k \\ _Charles R Greathouse IV_, Apr 28 2016

%Y Cf. A000203, A097018 (if n is a prime), A227470.

%K nonn

%O 1,2

%A _Waldemar Puszkarz_, Apr 26 2016