OFFSET
1,2
COMMENTS
Terms corresponding to prime indices p > 36 were calculated using the Giovanni Resta observation: a(p)=p*2^(p-1) for p>3.
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.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..500
EXAMPLE
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.
MATHEMATICA
PROG
(PARI) for(n=1, 44, k=n; while(!(k%n==0&&numdiv(k)%n==0), k+=n); print1(k ", ")) /* Puszkarz */
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Waldemar Puszkarz, Apr 26 2016
STATUS
approved