A100778 Integer powers of primorial numbers.

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

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

From Amiram Eldar, Sep 26 2023: (Start)

Intersection of A025487 and A072774.

The distinct terms of A046523(A072774(n)) in ascending orders.

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)

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, A046523, A055932, A056239, A057588, A072774, A072777, A112798, A304250, A319169, A322792, A322793.

Sequence in context: A068799 A317306 A317087 * A368325 A324579 A271520

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