Search: id:A001694
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%I A001694 M3325 N1335
%S A001694 1,4,8,9,16,25,27,32,36,49,64,72,81,100,108,121,125,128,144,169,196,
%T A001694 200,216,225,243,256,288,289,324,343,361,392,400,432,441,484,500,512,
%U A001694 529,576,625,648,675,676,729,784,800,841,864,900,961,968,972,1000
%N A001694 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).
%C A001694 In other words if the prime factorization of n is Product_k p_k^e_k then 
               all e_k are greater than 1.
%C A001694 n such that sum( d : n, phi(d)phi(n/d)mu(d)) > 0 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Nov 30 2002
%D A001694 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001694 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001694 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.
%D A001694 S. W. Golomb, Powerful numbers, Amer. Math. Monthly, 77 (1970), 848-852.
%D A001694 G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 
               37 (1983), 277-307.
%D A001694 A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
%D A001694 R. A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
%H A001694 T. D. Noe, <a href="b001694.txt">Table of n, a(n) for n = 1..1000</a>
%H A001694 <a href="Sindx_Pow.html#powerful">Index entries for sequences related 
               to powerful numbers</a>
%H A001694 C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=PowerfulNumber">
               Powerful Numbers</a>
%H A001694 K. Schneider, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               SquarefullNumber.html">squarefull number</a>
%H A001694 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PowerfulNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A001694 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Squareful.html">Link to a section of The World of Mathematics.</a>
%H A001694 Wikipedia, <a href="http://en.wikipedia.org/wiki/Powerful_number">Powerful 
               number</a>
%F A001694 Numbers of the form a^2*b^3, a>=1, b>= 1.
%t A001694 Select[ Range[ 2, 2500 ], Position[ Union[ Transpose[ FactorInteger[ 
               # ] ][ [ 2 ] ] ], 1 ] == {} & ]
%Y A001694 Cf. A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929.
%Y A001694 Cf. also A076871.
%Y A001694 Sequence in context: A109422 A158804 A080366 this_sequence A157985 A001597 
               A072777
%Y A001694 Adjacent sequences: A001691 A001692 A001693 this_sequence A001695 A001696 
               A001697
%K A001694 nonn,nice,easy
%O A001694 1,2
%A A001694 N. J. A. Sloane (njas(AT)research.att.com).
%E A001694 More terms from Henry Bottomley (se16(AT)btinternet.com), Mar 16 2000

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