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A048103
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Numbers not divisible by p^p for any prime p.
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69
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1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98
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OFFSET
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1,2
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COMMENTS
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If a(n) = Product p_i^e_i then p_i > e_i for all i.
Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. [Charles R Greathouse IV, Jan 25 2012]
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LINKS
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FORMULA
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a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. [Charles R Greathouse IV, Jan 25 2012]
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EXAMPLE
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6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.
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MATHEMATICA
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{1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)
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PROG
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(Haskell)
a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
(Scheme, with Antti Karttunen's IntSeq-library)
(PARI) isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i, 1] <= f[i, 2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020
(Python)
from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n:all(map(lambda d:d[1]<d[0], factorint(n).items())), count(max(startvalue, 1)))
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CROSSREFS
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Cf. A276086 (a permutation of this sequence).
Differs from its subsequence A276078 for the first time at n=451 where a(451)=625, while that value is missing from A276078.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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