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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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120
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
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
| 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
| 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, Powerful Number, Squareful
Wikipedia, Powerful number
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FORMULA
| Numbers of the form a^2*b^3, a>=1, b>= 1.
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MAPLE
| 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
| 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
| (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
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CROSSREFS
| Cf. A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929, A076871.
Sequence in context: A109422 A158804 A080366 * A157985 A001597 A072777
Adjacent sequences: A001691 A001692 A001693 * A001695 A001696 A001697
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Henry Bottomley (se16(AT)btinternet.com), Mar 16 2000
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