

A100778


Integer powers of primorial numbers.


13



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

1,2


COMMENTS

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


LINKS

David A. Corneth, Table of n, a(n) for n = 1..8606 (terms <= 10^1000)


FORMULA

Sum_{n>=1} 1/a(n) = 1 + Sum_{n>=1} 1/A057588(n) = 2.2397359032...  Amiram Eldar, Oct 20 2020; corrected by Hal M. Switkay and Amiram Eldar, Apr 12 2021


EXAMPLE

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.


MATHEMATICA

unintQ[n_]:=And[SameQ@@Last/@FactorInteger[n], Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[1, 1]]]];
Select[Range[1000], unintQ] (* Gus Wiseman, Dec 26 2018 *)


CROSSREFS

Cf. A000961, A001597, A002110, A007947, A025487, A055932, A056239, A057588, A072774, A072777, A112798, A304250, A319169, A322792, A322793.
Sequence in context: A068799 A317306 A317087 * A324579 A271520 A122408
Adjacent sequences: A100775 A100776 A100777 * A100779 A100780 A100781


KEYWORD

easy,nonn


AUTHOR

Amarnath Murthy, Nov 28 2004


EXTENSIONS

More terms and simpler definition from Ray Chandler, Nov 29 2004


STATUS

approved



