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 A324571 Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order. 16
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS These are a kind of self-describing numbers (cf. A001462, A304679). The increasing case is A109298. 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. LINKS EXAMPLE 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} MATHEMATICA Select[Range[1000], Reverse[PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]==Last/@If[#==1, {}, FactorInteger[#]]&] CROSSREFS Cf. A001156, A033461, A056239, A062457, A109298, A112798, A117144, A118914, A124010, A181819, A276078. Cf. A324524, A324525, A324572. Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360, A304679. Sequence in context: A253608 A126977 A102237 * A076505 A218073 A129345 Adjacent sequences:  A324568 A324569 A324570 * A324572 A324573 A324574 KEYWORD nonn AUTHOR Gus Wiseman, Mar 08 2019 STATUS approved

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Last modified September 26 11:41 EDT 2020. Contains 337368 sequences. (Running on oeis4.)