A322793 Proper powers of primorial numbers.

4, 8, 16, 32, 36, 64, 128, 216, 256, 512, 900, 1024, 1296, 2048, 4096, 7776, 8192, 16384, 27000, 32768, 44100, 46656, 65536, 131072, 262144, 279936, 524288, 810000, 1048576, 1679616, 2097152, 4194304, 5336100, 8388608, 9261000

Offset

1,1

Comments

A primorial number is a product of the first n primes, for some n.

Also Heinz numbers of non-strict 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).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..8255

Formula

Sum_{n>=1} 1/a(n) = Sum_{k>=1} 1/(A002110(k)*(A002110(k)-1)) = 0.53450573145072369022... . - Amiram Eldar, Mar 10 2024

Example

The sequence of all non-strict uniform integer partitions whose Heinz numbers belong to the sequence begins: (11), (111), (1111), (11111), (2211), (111111), (1111111), (222111), (11111111), (111111111), (332211), (1111111111), (22221111).

Mathematica

unintpropQ[n_]:=And[SameQ@@Last/@FactorInteger[n], FactorInteger[n][[1, 2]]>1, Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1, 1]]]];
Select[Range[10000], unintpropQ]
(* Second program: *)
nn = 2^24; k = 1; P = 2; Union@ Reap[While[j = 2; While[P^j < nn, Sow[P^j]; j++]; j > 2, k++; P *= Prime[k]]][[-1, 1]] (* Michael De Vlieger, Oct 04 2023 *)

Crossrefs

Cf. A000961, A001597, A001694, A002110, A025487, A047966, A055932, A056239, A072774, A072777, A100778, A112798, A304250, A322792.

Sequence in context: A354955 A304250 A368681 * A283052 A088259 A123857

Adjacent sequences: A322790 A322791 A322792 * A322794 A322795 A322796

Keyword

nonn

Author

Gus Wiseman, Dec 26 2018

Status

approved