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

Author

Gus Wiseman, Dec 26 2018

Keywords

• nonn

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).

Examples

			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).
		

Links

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

Crossrefs

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

Programs

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 *)

Formula

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