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A113849 Numbers whose prime factors are raised to the fourth power. 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
FORMULA
From Peter Munn, Oct 31 2019: (Start)
a(n) = A005117(n+1)^4.
{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
EXAMPLE
1296 = 16*81 = 2^4*3^4 so the prime factors of 1296, 2 and 3, are raised to the fourth power.
MATHEMATICA
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 *)
Rest[Select[Range[100], SquareFreeQ]^4] (* Vaclav Kotesovec, May 22 2020 *)
PROG
(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
CROSSREFS
Proper subset of A000583.
Other powers of squarefree numbers: A005117(1), A062503(2), A062838(3), A113850(5), A113851(6), A113852(7), A072774(all).
Sequence in context: A153157 A369168 A366307 * A046453 A030514 A056571
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Jan 25 2006
EXTENSIONS
More terms from Robert G. Wilson v, Jan 26 2006
STATUS
approved

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Last modified March 29 02:16 EDT 2024. Contains 371264 sequences. (Running on oeis4.)