A272348 - OEIS

%I #28 Apr 28 2016 11:30:46

%S 1,2,9,8,80,12,448,24,36,80,11264,60,53248,448,720,384,1114112,180,

%T 4980736,240,4032,11264,96468992,360,10000,53248,24300,1344,

%U 7784628224,720,33285996544,1920,101376,1114112,181440,1260,2542620639232,4980736,479232,1680,45079976738816,4032,189115999977472,33792

%N Least number divisible by n whose number of divisors is also divisible by n.

%C Terms corresponding to prime indices p > 36 were calculated using the _Giovanni Resta_ observation: a(p)=p*2^(p-1) for p>3.

%C Proof. There are two candidates for terms corresponding to prime indices p: p*2^(p-1) and p^(p-1). Both have the number of divisors divisible by p, as in the first case it is 2p and in the other p. Both are also divisible by p. For 2 and 3, p^(p-1) < p*2^(p-1) and so we have a(2)=2^1=2 and a(3)=3^2=9. However, this is no longer the case for primes greater than 3. For instance, 5^4=625 > 5*2^4=80, hence a(5)=80. The same holds true for other primes.

%H Giovanni Resta, <a href="/A272348/b272348.txt">Table of n, a(n) for n = 1..500</a>

%e For n=3, a(3)=9 because it is the smallest number divisible by 3 whose number of divisors (3) is also divisible by 3; numbers less than 9 either are not divisible by 3 (1,2,4,5,7,8) or their number of divisors is not (3,6) being (2,4), respectively.

%t A272348 = {}; Do[k = n; While[!(Divisible[k, n] && Divisible[DivisorSigma[0, k], n]), k++]; AppendTo[A272348, k], {n, 44}]; lst (* Puszkarz *)

%t A272348 = {}; Do[k = n; If[(n < 4) || CompositeQ[n], While[!(Divisible[k, n] && Divisible[DivisorSigma[0, k], n]), k++]; AppendTo[A272348, k], AppendTo[A272348, n*2^(n - 1)]], {n, 44}]; A272348 (* uses a(p)=p*2^(p-1) for p>3 prime *)

%o (PARI) for(n=1, 44, k=n; while(!(k%n==0&&numdiv(k)%n==0), k+=n); print1(k ", ")) /* Puszkarz */

%o (PARI) for(n=1,44, k=n; if(n<4||!isprime(n), while(!(k%n==0&&numdiv(k)%n==0), k+=n), k=n*2^(n-1)); print1(k, ", ")) /* uses a(p)=p*2^(p-1) for p>3 prime */

%Y Cf. A000005 (number of divisors).

%K nonn

%O 1,2

%A _Waldemar Puszkarz_, Apr 26 2016