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A113849
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Numbers whose prime factors are raised to the fourth power.
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20
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16, 81, 625, 1296, 2401, 10000, 14641, 28561, 38416, 50625, 83521, 130321, 194481, 234256, 279841, 456976, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1874161, 2085136, 2313441, 2825761, 3111696, 3418801, 4477456, 4879681, 6765201, 7890481
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OFFSET
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1,1
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COMMENTS
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This is essentially A005117 (the squarefree numbers) raised to the fourth power. - T. D. Noe, Mar 13 2013
All positive integers have a unique factorization into powers of squarefree numbers with distinct exponents that are powers of two. So every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (term of this sequence), at most one 8th power of a squarefree number, and so on. - Peter Munn, Mar 12 2020
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LINKS
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FORMULA
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{a(n)} = {A225546(A000351(n)) : n >= 0} \ {1}, where {a(n)} denotes the set of integers in the sequence.
(End)
Sum_{k>=1} 1/a(k) = zeta(4)/zeta(8) - 1 = 105/Pi^4 - 1. - Amiram Eldar, May 22 2020
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EXAMPLE
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1296 = 16*81 = 2^4*3^4 so the prime factors of 1296, 2 and 3, are raised to the fourth power.
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MATHEMATICA
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Select[ Range@50^4, Union[Last /@ FactorInteger@# ] == {4} &] (* Robert G. Wilson v, Jan 26 2006 *)
nn = 50; t = Select[Range[2, nn], Union[Transpose[FactorInteger[#]][[2]]] == {1} &]; t^4 (* T. D. Noe, Mar 13 2013 *)
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PROG
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(PARI) allpwrfact(n, p) = { local(x, j, ln, y, flag); for(x=4, n, y=Vec(factor(x)); ln = length(y[1]); flag=0; for(j=1, ln, if(y[2][j]==p, flag++); ); if(flag==ln, print1(x", ")); ) } \\ All prime factors are raised to the power p
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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