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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. 16
 1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 162, 250, 256, 288, 324, 500, 512, 576, 648, 729, 1000, 1024, 1125, 1152, 1296, 1458, 2000, 2048, 2250, 2304, 2401, 2592, 2916, 4000, 4096, 4500, 4608, 4802, 5184, 5832, 6561, 8000, 8192, 9000, 9216 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS These are a kind of self-describing numbers (cf. A001462, A304679). 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. 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). Also products of elements of A062457. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 FORMULA Closed under multiplication. Sum_{n>=1} 1/a(n) = Product_{k>=1} 1/(1-prime(k)^(-k)) = 2.26910478689594012492... - Amiram Eldar, Sep 30 2020 EXAMPLE 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).     1: {}     2: {1}     4: {1,1}     8: {1,1,1}     9: {2,2}    16: {1,1,1,1}    18: {1,2,2}    32: {1,1,1,1,1}    36: {1,1,2,2}    64: {1,1,1,1,1,1}    72: {1,1,1,2,2}    81: {2,2,2,2}   125: {3,3,3}   128: {1,1,1,1,1,1,1}   144: {1,1,1,1,2,2}   162: {1,2,2,2,2}   250: {1,3,3,3}   256: {1,1,1,1,1,1,1,1} MAPLE q:= n-> andmap(i-> irem(i[2], numtheory[pi](i[1]))=0, ifactors(n)[2]): select(q, [\$1..10000])[];  # Alois P. Heinz, Mar 08 2019 MATHEMATICA Select[Range[1000], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>Divisible[k, PrimePi[p]]]&] 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 *) CROSSREFS Cf. A001156, A033461, A056239, A062457, A066328, A072873, A112798, A118914 (prime signature), A124010, A181819, A276078, A304679. Cf. A109298, A324525, A324570, A324571, A324572, A324587, A324588. Range of values of A090884. Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360. Sequence in context: A080025 A152111 A316856 * A325621 A025611 A049439 Adjacent sequences:  A324521 A324522 A324523 * A324525 A324526 A324527 KEYWORD nonn,changed AUTHOR Gus Wiseman, Mar 07 2019 STATUS approved

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Last modified August 16 04:07 EDT 2022. Contains 356150 sequences. (Running on oeis4.)