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A272348
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Least number divisible by n whose number of divisors is also divisible by n.
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1
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1, 2, 9, 8, 80, 12, 448, 24, 36, 80, 11264, 60, 53248, 448, 720, 384, 1114112, 180, 4980736, 240, 4032, 11264, 96468992, 360, 10000, 53248, 24300, 1344, 7784628224, 720, 33285996544, 1920, 101376, 1114112, 181440, 1260, 2542620639232, 4980736, 479232, 1680, 45079976738816, 4032, 189115999977472, 33792
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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A272348 = {}; Do[k = n; While[!(Divisible[k, n] && Divisible[DivisorSigma[0, k], n]), k++]; AppendTo[A272348, k], {n, 44}]; lst (* Puszkarz *)
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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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