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A325127 Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.

%I A325127 #12 Oct 24 2020 04:12:41
%S A325127 1,4,8,16,27,32,64,81,108,128,216,243,256,324,432,512,625,648,729,864,
%T A325127 972,1024,1296,1728,1944,2048,2187,2500,2592,2916,3125,3456,3888,4096,
%U A325127 5000,5184,5832,6561,6912,7776,8192,8748,10000,10368,11664,12500,13824,15552
%N A325127 Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.
%C A325127 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A325127 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions where each part k appears more than k times. Such partitions are counted by A115584.
%H A325127 Amiram Eldar, <a href="/A325127/b325127.txt">Table of n, a(n) for n = 1..2847</a> (terms up to 10^12)
%F A325127 Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^k * (prime(k)-1)) = 1.58661114052385082598.... - _Amiram Eldar_, Oct 24 2020
%e A325127 The sequence of terms together with their prime indices begins:
%e A325127     1: {}
%e A325127     4: {1,1}
%e A325127     8: {1,1,1}
%e A325127    16: {1,1,1,1}
%e A325127    27: {2,2,2}
%e A325127    32: {1,1,1,1,1}
%e A325127    64: {1,1,1,1,1,1}
%e A325127    81: {2,2,2,2}
%e A325127   108: {1,1,2,2,2}
%e A325127   128: {1,1,1,1,1,1,1}
%e A325127   216: {1,1,1,2,2,2}
%e A325127   243: {2,2,2,2,2}
%e A325127   256: {1,1,1,1,1,1,1,1}
%e A325127   324: {1,1,2,2,2,2}
%e A325127   432: {1,1,1,1,2,2,2}
%e A325127   512: {1,1,1,1,1,1,1,1,1}
%e A325127   625: {3,3,3,3}
%e A325127   648: {1,1,1,2,2,2,2}
%e A325127   729: {2,2,2,2,2,2}
%e A325127   864: {1,1,1,1,1,2,2,2}
%e A325127   972: {1,1,2,2,2,2,2}
%t A325127 Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>PrimePi[p]]&]
%t A325127 With[{k = 4}, m = Prime[k]^(k + 1); s = {}; Do[p = Prime[i]; AppendTo[s, Join[{1}, p^Range[i + 1, Floor[Log[p, m]]]]], {i, 1, k}]; Union @ Select[Times @@@ Tuples[s], # <= m &]] (* _Amiram Eldar_, Oct 24 2020 *)
%Y A325127 Cf. A000720, A056239, A062457, A109298, A112798, A115584, A118914, A276078.
%Y A325127 Cf. A324524, A324525, A324571, A325128, A325130.
%K A325127 nonn
%O A325127 1,2
%A A325127 _Gus Wiseman_, Apr 01 2019