login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
139
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)
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.

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

a(n) = A224866(n) - 1. - Reinhard Zumkeller, Jul 23 2013

The number of terms less than or equal to 10^k beginning with k=0: 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, ...: A118896. - Robert G. Wilson v, Aug 11 2014

a(10^n): 1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, …, . - Robert G. Wilson v, Aug 15 2014

REFERENCES

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

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

P. Erdos and G. Szekeres, Über 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, Vol. 77 (1970), 848-852.

K. Schneider, PlanetMath.org, Squarefull Number, Powerful Number, Squareful

Wikipedia, Powerful number

Index entries for sequences related to powerful numbers

FORMULA

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

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

Join[{1}, Select[ Range@ 1250, Min@ FactorInteger[#][[All, 2]] > 1 &]]

(* Harvey P. Dale, Sep 18 2011 and modified by Robert G. Wilson v, Aug 11 2014 *)

max = 10^3; Union@ Flatten@ Table[a^2*b^3, {b, max^(1/3)}, {a, Sqrt[max/b^3]}] (* Robert G. Wilson v, Aug 11 2014 *)

nextPowerfulNumber[n_] := Block[{r = Range[ Floor[1 + n^(1/3)]]^3}, Min@ Select[ Sort[ r*Floor[1 + Sqrt[n/r]]^2], # > n &]]; NestList[ nextPowerfulNumber, 1, 55] (* Robert G. Wilson v, Aug 16 2014 *)

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

(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

(Python)

from sympy import factorint

A001694 = [1]+[n for n in xrange(2, 10**6) if min(factorint(n).values()) > 1]

# Chai Wah Wu, Aug 14 2014

CROSSREFS

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

KEYWORD

nonn,nice,easy,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Henry Bottomley, Mar 16 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified August 30 20:23 EDT 2014. Contains 246229 sequences.