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A054744 p-full numbers: numbers such that if any prime p divides it, then so does p^p.

%I A054744 #17 Jul 02 2025 16:01:59
%S A054744 1,4,8,16,27,32,64,81,108,128,216,243,256,324,432,512,648,729,864,972,
%T A054744 1024,1296,1728,1944,2048,2187,2592,2916,3125,3456,3888,4096,5184,
%U A054744 5832,6561,6912,7776,8192,8748,10368,11664,12500,13824,15552,15625,16384
%N A054744 p-full numbers: numbers such that if any prime p divides it, then so does p^p.
%C A054744 A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). [_Reinhard Zumkeller_, Apr 28 2012]
%C A054744 Heinz numbers of integer partitions where the multiplicity of each part k is at least prime(k). These partitions are counted by A325132. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - _Gus Wiseman_, Apr 02 2019
%H A054744 Donovan Johnson, <a href="/A054744/b054744.txt">Table of n, a(n) for n = 1..10000</a>
%F A054744 If n = Product p_i^e_i then p_i<=e_i for all i.
%F A054744 Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/(p^(p-1)*(p-1)) = 1.58396891058853238595.... - _Amiram Eldar_, Oct 24 2020
%e A054744 8 is an element because 8 = 2^3 and 2<=3, while 25 is not an element because 25 = 5^2 and 5>2.
%e A054744 From _Gus Wiseman_, Apr 02 2019: (Start)
%e A054744 The sequence of terms together with their prime indices begins:
%e A054744     1: {}
%e A054744     4: {1,1}
%e A054744     8: {1,1,1}
%e A054744    16: {1,1,1,1}
%e A054744    27: {2,2,2}
%e A054744    32: {1,1,1,1,1}
%e A054744    64: {1,1,1,1,1,1}
%e A054744    81: {2,2,2,2}
%e A054744   108: {1,1,2,2,2}
%e A054744   128: {1,1,1,1,1,1,1}
%e A054744   216: {1,1,1,2,2,2}
%e A054744   243: {2,2,2,2,2}
%e A054744   256: {1,1,1,1,1,1,1,1}
%e A054744   324: {1,1,2,2,2,2}
%e A054744   432: {1,1,1,1,2,2,2}
%e A054744   512: {1,1,1,1,1,1,1,1,1}
%e A054744   648: {1,1,1,2,2,2,2}
%e A054744   729: {2,2,2,2,2,2}
%e A054744   864: {1,1,1,1,1,2,2,2}
%e A054744   972: {1,1,2,2,2,2,2}
%e A054744 (End)
%t A054744 Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=p]&] (* _Gus Wiseman_, Apr 02 2019 *)
%o A054744 (Haskell)
%o A054744 a054744 n = a054744_list !! (n-1)
%o A054744 a054744_list = filter (\x -> and $
%o A054744    zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
%o A054744 -- _Reinhard Zumkeller_, Apr 28 2012
%Y A054744 Cf. A048103, A001694, A036966.
%Y A054744 Cf. A100716.
%Y A054744 Cf. A056239, A109298, A112798, A115584, A117144, A124010, A276078, A324525, A324571, A325127, A325128, A325130, A325131, A325132.
%K A054744 easy,nonn
%O A054744 1,2
%A A054744 _James Sellers_, Apr 22 2000