|
|
A322793
|
|
Proper powers of primorial numbers.
|
|
4
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
|
|
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.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|