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A325359
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Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.
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8
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9, 18, 25, 27, 36, 49, 50, 54, 72, 81, 98, 100, 108, 121, 125, 144, 162, 169, 196, 200, 216, 242, 243, 250, 288, 289, 324, 338, 343, 361, 392, 400, 432, 484, 486, 500, 529, 576, 578, 625, 648, 676, 686, 722, 729, 784, 800, 841, 864, 961, 968, 972, 1000, 1058
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OFFSET
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1,1
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COMMENTS
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Also Heinz numbers of integer partitions that are not hooks but whose augmented differences are hooks, where the Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k), and a hook is a partition of the form (n,1,1,...,1). The enumeration of these partitions by sum is given by A325459.
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = 2 * Sum_{p prime} 1/(p*(p-1)) - 1 = 2 * A136141 - 1 = 0.54631333809959025572... - Amiram Eldar, Sep 30 2020
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EXAMPLE
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The sequence of terms together with their prime indices begins:
9: {2,2}
18: {1,2,2}
25: {3,3}
27: {2,2,2}
36: {1,1,2,2}
49: {4,4}
50: {1,3,3}
54: {1,2,2,2}
72: {1,1,1,2,2}
81: {2,2,2,2}
98: {1,4,4}
100: {1,1,3,3}
108: {1,1,2,2,2}
121: {5,5}
125: {3,3,3}
144: {1,1,1,1,2,2}
162: {1,2,2,2,2}
169: {6,6}
196: {1,1,4,4}
200: {1,1,1,3,3}
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MAPLE
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N:= 1000: # to get terms <= N
P:= select(isprime, [seq(i, i=3..floor(sqrt(N)), 2)]):
B:= map(proc(p) local y; seq(p^y, y=2..floor(log[p](N))) end proc, P):
sort(map(proc(t) local z; seq(2^z*t, z=0..ilog2(N/t)) end proc, B)); # Robert Israel, May 03 2019
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MATHEMATICA
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Select[Range[1000], MatchQ[FactorInteger[2*#], {{2, _}, {_?(#>2&), _?(#>1&)}}]&]
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CROSSREFS
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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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