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A324571
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Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.
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16
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1, 2, 9, 12, 40, 112, 125, 352, 360, 675, 832, 1008, 2176, 2401, 3168, 3969, 4864, 7488, 11776, 14000, 19584, 29403, 29696, 43776, 44000, 63488, 75600, 104000, 105984, 123201, 151552, 161051, 214375, 237600, 267264, 272000, 335872, 496125, 561600, 571392, 608000
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OFFSET
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1,2
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COMMENTS
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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.
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).
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.
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LINKS
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EXAMPLE
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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).
1: {}
2: {1}
9: {2,2}
12: {1,1,2}
40: {1,1,1,3}
112: {1,1,1,1,4}
125: {3,3,3}
352: {1,1,1,1,1,5}
360: {1,1,1,2,2,3}
675: {2,2,2,3,3}
832: {1,1,1,1,1,1,6}
1008: {1,1,1,1,2,2,4}
2176: {1,1,1,1,1,1,1,7}
2401: {4,4,4,4}
3168: {1,1,1,1,1,2,2,5}
3969: {2,2,2,2,4,4}
4864: {1,1,1,1,1,1,1,1,8}
7488: {1,1,1,1,1,1,2,2,6}
11776: {1,1,1,1,1,1,1,1,1,9}
14000: {1,1,1,1,3,3,3,4}
19584: {1,1,1,1,1,1,1,2,2,7}
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MATHEMATICA
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Select[Range[1000], Reverse[PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]==Last/@If[#==1, {}, FactorInteger[#]]&]
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CROSSREFS
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Cf. A001156, A033461, A056239, A062457, A109298, A112798, A117144, A118914, A124010, A181819, A276078.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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