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%I #10 Mar 10 2019 20:59:28
%S 1,2,9,12,40,112,125,352,360,675,832,1008,2176,2401,3168,3969,4864,
%T 7488,11776,14000,19584,29403,29696,43776,44000,63488,75600,104000,
%U 105984,123201,151552,161051,214375,237600,267264,272000,335872,496125,561600,571392,608000
%N Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.
%C These are a kind of self-describing numbers (cf. A001462, A304679). The increasing case is A109298.
%C 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 ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base.
%C Also Heinz numbers of the integer partitions counted by A324572. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C Each finite set of positive integers determines a unique term with those prime indices. For example, corresponding to {1,2,4,5} is 1397088 = prime(1)^5 * prime(2)^4 * prime(4)^2 * prime(5)^1.
%e The sequence of terms together with their prime indices begins as follows. For example, we have 40: {1,1,1,3} because 40 = prime(1) * prime(1) * prime(1) * prime(3).
%e 1: {}
%e 2: {1}
%e 9: {2,2}
%e 12: {1,1,2}
%e 40: {1,1,1,3}
%e 112: {1,1,1,1,4}
%e 125: {3,3,3}
%e 352: {1,1,1,1,1,5}
%e 360: {1,1,1,2,2,3}
%e 675: {2,2,2,3,3}
%e 832: {1,1,1,1,1,1,6}
%e 1008: {1,1,1,1,2,2,4}
%e 2176: {1,1,1,1,1,1,1,7}
%e 2401: {4,4,4,4}
%e 3168: {1,1,1,1,1,2,2,5}
%e 3969: {2,2,2,2,4,4}
%e 4864: {1,1,1,1,1,1,1,1,8}
%e 7488: {1,1,1,1,1,1,2,2,6}
%e 11776: {1,1,1,1,1,1,1,1,1,9}
%e 14000: {1,1,1,1,3,3,3,4}
%e 19584: {1,1,1,1,1,1,1,2,2,7}
%t Select[Range[1000],Reverse[PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==Last/@If[#==1,{},FactorInteger[#]]&]
%Y Cf. A001156, A033461, A056239, A062457, A109298, A112798, A117144, A118914, A124010, A181819, A276078.
%Y Cf. A324524, A324525, A324572.
%Y Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360, A304679.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 08 2019
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