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A054744
p-full numbers: numbers such that if any prime p divides it, then so does p^p.
11
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
OFFSET
1,2
COMMENTS
A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). [Reinhard Zumkeller, Apr 28 2012]
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
LINKS
FORMULA
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
EXAMPLE
8 is an element because 8 = 2^3 and 2<=3, while 25 is not an element because 25 = 5^2 and 5>2.
From Gus Wiseman, Apr 02 2019: (Start)
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)
MATHEMATICA
Select[Range[1000], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k>=p]&] (* Gus Wiseman, Apr 02 2019 *)
PROG
(Haskell)
a054744 n = a054744_list !! (n-1)
a054744_list = filter (\x -> and $
zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012
KEYWORD
easy,nonn
AUTHOR
James A. Sellers, Apr 22 2000
STATUS
approved