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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).
(Formerly M3325 N1335)
120
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)
OFFSET

1,2

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]

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).

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

FORMULA

Numbers of the form a^2*b^3, a>=1, b>= 1.

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

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 *)

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

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

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com).

EXTENSIONS

More terms from Henry Bottomley (se16(AT)btinternet.com), Mar 16 2000

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Last modified February 23 03:36 EST 2012. Contains 206606 sequences.