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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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127
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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;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
Numbers n such that sum( d | n, phi(d)*phi(n/d)*mu(d) ) > 0 - Benoit Cloitre, Nov 30 2002
This sequence is closed under multiplication. The primitive elements are A168363. - Franklin T. Adams-Watters, May 30, 2011.
Complement of A052485; A112526(a(n)) = 1. [Reinhard Zumkeller, Sep 16 2011]
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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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98.
C. K. Caldwell, Powerful Numbers
K. Schneider, PlanetMath.org, Squarefull Number, Powerful Number, Squareful
Wikipedia, Powerful number
Index entries for sequences related to powerful numbers
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FORMULA
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Numbers of the form a^2*b^3, a>=1, b>= 1.
Bateman & Grosswald prove that a(n) = zeta(3/2)/zeta(3) x^{1/2} + zeta(2/3)/zeta(2) x^{1/3} + O(x^{1/6}); see section 5 for a more precise error term. - Charles R Greathouse IV, Nov 19 2012
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MAPLE
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isA001694 := proc(n) for p in ifactors(n)[2] do if op(2, p) = 1 then return false; end if; end do; return true; end proc:
A001694 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001694(a) then return a; end if; end do; end if; end proc:
seq(A001694(n), n=1..20) ; # R. J. Mathar, Jun 07 2011
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MATHEMATICA
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Select[ Range[ 2, 2500 ], Position[ Union[ Transpose[ FactorInteger[ # ] ][ [ 2 ] ] ], 1 ] == {} & ]
Join[{1}, Select[Range[1000], Min[Transpose[FactorInteger[#]] [[2]]]>1&]] (* From Harvey P. Dale, Sep 18 2011 *)
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PROG
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(PARI) isA001694(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]==1, return(0))); 1 \\ Charles R Greathouse IV, Feb 11 2011
(PARI) list(lim, mn=2)=my(v=List(), t); for(m=1, lim^(1/3), t=m^3; for(n=1, sqrtint(lim\t), listput(v, t*n^2))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Jul 31 2011
(PARI) is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012
(Haskell)
a001694 n = a001694_list !! (n-1)
a001694_list = filter ((== 1) . a112526) [1..]
-- Reinhard Zumkeller, Nov 30 2012
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CROSSREFS
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Cf. A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929, A076871.
Cf. A076446 (first differences).
Sequence in context: A109422 A158804 A080366 * A157985 A001597 A072777
Adjacent sequences: A001691 A001692 A001693 * A001695 A001696 A001697
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KEYWORD
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nonn,nice,easy,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Henry Bottomley, Mar 16 2000
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STATUS
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approved
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