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A100778
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Integer powers of primorial numbers.
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15
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1, 2, 4, 6, 8, 16, 30, 32, 36, 64, 128, 210, 216, 256, 512, 900, 1024, 1296, 2048, 2310, 4096, 7776, 8192, 16384, 27000, 30030, 32768, 44100, 46656, 65536, 131072, 262144, 279936, 510510, 524288, 810000, 1048576, 1679616, 2097152, 4194304, 5336100
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OFFSET
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1,2
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COMMENTS
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Smallest squarefree numbers or their powers with distinct prime signatures. Or least numbers with prime signatures (p*q*r*...)^k, where p,q,r,... are primes and k is a whole number.
Also Heinz numbers of uniform integer partitions whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all uniform integer partitions whose Heinz numbers belong to the sequence begins: (1), (11), (12), (111), (1111), (123), (11111), (1122), (111111), (1111111), (1234), (111222), (11111111), (111111111), (112233), (1111111111). - Gus Wiseman, Dec 26 2018
The k-th power of the n-th primorial number, A002110(n)^k, has (k+1)^n divisors which are the set of the (k+1)-free prime(n)-smooth numbers. (End)
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LINKS
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FORMULA
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EXAMPLE
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10 is not a term as 6 is a member with the same prime signature 10 > 6.
216 is a term as 216 = (2*3)^3. 243 is not a term as 32 represents that prime signature.
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MATHEMATICA
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unintQ[n_]:=And[SameQ@@Last/@FactorInteger[n], Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1, 1]]]];
Select[Range[1000], unintQ] (* Gus Wiseman, Dec 26 2018 *)
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CROSSREFS
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Cf. A000961, A001597, A002110, A007947, A025487, A046523, A055932, A056239, A057588, A072774, A072777, A112798, A304250, A319169, A322792, A322793.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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More terms and simpler definition from Ray Chandler, Nov 29 2004
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STATUS
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approved
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