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Proper powers of primorial numbers.
4

%I #15 Mar 10 2024 04:02:08

%S 4,8,16,32,36,64,128,216,256,512,900,1024,1296,2048,4096,7776,8192,

%T 16384,27000,32768,44100,46656,65536,131072,262144,279936,524288,

%U 810000,1048576,1679616,2097152,4194304,5336100,8388608,9261000

%N Proper powers of primorial numbers.

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

%C 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).

%H Michael De Vlieger, <a href="/A322793/b322793.txt">Table of n, a(n) for n = 1..8255</a>

%F Sum_{n>=1} 1/a(n) = Sum_{k>=1} 1/(A002110(k)*(A002110(k)-1)) = 0.53450573145072369022... . - _Amiram Eldar_, Mar 10 2024

%e 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).

%t unintpropQ[n_]:=And[SameQ@@Last/@FactorInteger[n],FactorInteger[n][[1,2]]>1,Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1,1]]]];

%t Select[Range[10000],unintpropQ]

%t (* Second program: *)

%t 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 *)

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 26 2018