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Search: id:A001694
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| A001694 |
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Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squarefull, square-full or 2-full numbers).
(Formerly M3325 N1335)
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+0
96
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| 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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In other words if the prime factorization of n is Product_k p_k^e_k then all e_k are greater than 1.
n such that sum( d : n, phi(d)phi(n/d)mu(d)) > 0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 30 2002
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
P. Erdos and G. Szekeres, Ueber die Anzahl der Abelschen Gruppen gegebener Ordnung und ueber ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
S. W. Golomb, Powerful numbers, Amer. Math. Monthly, 77 (1970), 848-852.
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.
A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
R. A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for sequences related to powerful numbers
C. K. Caldwell, Powerful Numbers
K. Schneider, PlanetMath.org, squarefull number
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Wikipedia, Powerful number
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FORMULA
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Numbers of the form a^2*b^3, a>=1, b>= 1.
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MATHEMATICA
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Select[ Range[ 2, 2500 ], Position[ Union[ Transpose[ FactorInteger[ # ] ][ [ 2 ] ] ], 1 ] == {} & ]
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CROSSREFS
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Cf. A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929.
Cf. also A076871.
Sequence in context: A109422 A158804 A080366 this_sequence A157985 A001597 A072777
Adjacent sequences: A001691 A001692 A001693 this_sequence A001695 A001696 A001697
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Henry Bottomley (se16(AT)btinternet.com), Mar 16 2000
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