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A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

%I A324524 #27 Aug 04 2022 15:09:20
%S A324524 1,2,4,8,9,16,18,32,36,64,72,81,125,128,144,162,250,256,288,324,500,
%T A324524 512,576,648,729,1000,1024,1125,1152,1296,1458,2000,2048,2250,2304,
%U A324524 2401,2592,2916,4000,4096,4500,4608,4802,5184,5832,6561,8000,8192,9000,9216
%N A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.
%C A324524 These are a kind of self-describing numbers (cf. A001462, A304679).
%C A324524 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. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
%C A324524 Also Heinz numbers of integer partitions in which every part divides its multiplicity (counted by A001156). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A324524 Also products of elements of A062457.
%H A324524 Amiram Eldar, <a href="/A324524/b324524.txt">Table of n, a(n) for n = 1..10000</a>
%F A324524 Closed under multiplication.
%F A324524 Sum_{n>=1} 1/a(n) = Product_{k>=1} 1/(1-prime(k)^(-k)) = 2.26910478689594012492... - _Amiram Eldar_, Sep 30 2020
%e A324524 The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2).
%e A324524     1: {}
%e A324524     2: {1}
%e A324524     4: {1,1}
%e A324524     8: {1,1,1}
%e A324524     9: {2,2}
%e A324524    16: {1,1,1,1}
%e A324524    18: {1,2,2}
%e A324524    32: {1,1,1,1,1}
%e A324524    36: {1,1,2,2}
%e A324524    64: {1,1,1,1,1,1}
%e A324524    72: {1,1,1,2,2}
%e A324524    81: {2,2,2,2}
%e A324524   125: {3,3,3}
%e A324524   128: {1,1,1,1,1,1,1}
%e A324524   144: {1,1,1,1,2,2}
%e A324524   162: {1,2,2,2,2}
%e A324524   250: {1,3,3,3}
%e A324524   256: {1,1,1,1,1,1,1,1}
%p A324524 q:= n-> andmap(i-> irem(i[2], numtheory[pi](i[1]))=0, ifactors(n)[2]):
%p A324524 select(q, [$1..10000])[];  # _Alois P. Heinz_, Mar 08 2019
%t A324524 Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[k,PrimePi[p]]]&]
%t A324524 v = Join[{1}, Prime[(r = Range[10])]^r]; n = Length[v]; vmax = 10^4; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; Length[s] (* _Amiram Eldar_, Sep 30 2020 *)
%Y A324524 Cf. A001156, A033461, A056239, A062457, A066328, A072873, A112798, A118914 (prime signature), A124010, A181819, A276078, A304679.
%Y A324524 Cf. A109298, A324525, A324570, A324571, A324572, A324587, A324588.
%Y A324524 Range of values of A090884.
%Y A324524 Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.
%K A324524 nonn
%O A324524 1,2
%A A324524 _Gus Wiseman_, Mar 07 2019