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A054744
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p-full numbers: numbers such that if any prime p divides it, then so does p^p.
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11
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1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2592, 2916, 3125, 3456, 3888, 4096, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10368, 11664, 12500, 13824, 15552, 15625, 16384
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OFFSET
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1,2
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COMMENTS
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Heinz numbers of integer partitions where the multiplicity of each part k is at least prime(k). These partitions are counted by A325132. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 02 2019
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LINKS
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FORMULA
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If n = Product p_i^e_i then p_i<=e_i for all i.
Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/(p^(p-1)*(p-1)) = 1.58396891058853238595.... - Amiram Eldar, Oct 24 2020
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EXAMPLE
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8 is an element because 8 = 2^3 and 2<=3, while 25 is not an element because 25 = 5^2 and 5>2.
The sequence of terms together with their prime indices begins:
1: {}
4: {1,1}
8: {1,1,1}
16: {1,1,1,1}
27: {2,2,2}
32: {1,1,1,1,1}
64: {1,1,1,1,1,1}
81: {2,2,2,2}
108: {1,1,2,2,2}
128: {1,1,1,1,1,1,1}
216: {1,1,1,2,2,2}
243: {2,2,2,2,2}
256: {1,1,1,1,1,1,1,1}
324: {1,1,2,2,2,2}
432: {1,1,1,1,2,2,2}
512: {1,1,1,1,1,1,1,1,1}
648: {1,1,1,2,2,2,2}
729: {2,2,2,2,2,2}
864: {1,1,1,1,1,2,2,2}
972: {1,1,2,2,2,2,2}
(End)
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MATHEMATICA
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Select[Range[1000], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k>=p]&] (* Gus Wiseman, Apr 02 2019 *)
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PROG
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(Haskell)
a054744 n = a054744_list !! (n-1)
a054744_list = filter (\x -> and $
zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
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CROSSREFS
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Cf. A056239, A109298, A112798, A115584, A117144, A124010, A276078, A324525, A324571, A325127, A325128, A325130, A325131, A325132.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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