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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
90
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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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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
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: A127398 A109422 A080366 this_sequence A001597 A072777 A076292
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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njas
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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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