OFFSET
1,1
COMMENTS
"Erdős conjectured in 1975 that there do not exist three consecutive powerful integers." - Guy
See Guy for Erdős's conjecture and statement that this sequence is infinite. - Jud McCranie, Oct 13 2002
It is easy to see that this sequence is infinite: if k is in the sequence, so is 4*k*(k+1). - Franklin T. Adams-Watters, Sep 16 2009
The first of a run of three consecutive powerful numbers (conjectured to be empty) are just those in this sequence and A076445. - Charles R Greathouse IV, Nov 16 2012
Jaroslaw Wroblewski (see Prime Puzzles link) shows that there are infinitely many terms k in this sequence such that neither k nor k+1 is a square. - Charles R Greathouse IV, Nov 19 2012
Paul Erdős wrote of meeting Kurt Mahler in 1936: "I almost immediately posed him the following problem: ... are there infinitely many consecutive powerful numbers? Mahler immediately answered: Trivially, yes! x^2 - 8y^2 = 1 has infinitely many solutions. I was a bit crestfallen since I felt that I should have thought of this myself." - Jonathan Sondow, Feb 08 2015
Of the first 39 terms k, only 7 are such that neither k nor k+1 is a square. - Jon E. Schoenfield, Jun 12 2024
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 288, pp. 74, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B16.
P. Shiu, On the number of square-full integers between successive squares, Volume 27, Issue 2 (December 1980), pp. 171-178.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..39 (terms < 10^22)
C. K. Caldwell, Powerful Numbers.
P. Erdős, Some personal and mathematical reminiscences of Kurt Mahler, Austral. Math. Soc. Gaz., 16 (1) (1989), 1-2.
Jérôme Germoni, Nombres puissants au bac S, Images des Mathématiques, CNRS, 2018 (in French).
J. J. O'Connor and E. F. Robertson, Biography of Kurt Mahler.
Carlos Rivera, Problem 53. Powerful numbers revisited, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Powerful numbers.
EXAMPLE
1825200 belongs to this sequence because both 1825200 = 2^4 * 3^3 * 5^2 * 13^2 and 1825201 = 7^2 * 193^2 = 1351^2 are powerful numbers. - Labos Elemer, May 03 2001
MATHEMATICA
f[n_]:=First[Union[Last/@FactorInteger[n]]]; Select[Range[2000000], f[#]>1&&f[#+1]>1&] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
SequencePosition[Table[If[Min[FactorInteger[n][[;; , 2]]]>1, 1, 0], {n, 11310000}], {1, 1}][[;; , 1]] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Mar 27 2024 *)
PROG
(PARI) is(n)=ispowerful(n)&&ispowerful(n+1) \\ Charles R Greathouse IV, Nov 16 2012
(Haskell)
a060355 n = a060355_list !! (n-1)
a060355_list = map a001694 $ filter ((== 1) . a076446) [1..]
-- Reinhard Zumkeller, Jun 03 2015, Nov 30 2012
(Sage)
def A060355(n):
a = sloane.A001694
return a.is_powerful(n) and a.is_powerful(n+1)
[n for n in (1..333333) if A060355(n)] # Peter Luschny, Feb 08 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Jason Earls, Apr 01 2001
EXTENSIONS
Corrected and extended by Jud McCranie, Jul 08 2001
More terms from Jud McCranie, Oct 13 2002
a(22)-a(23) from Donovan Johnson, Jul 29 2011
STATUS
approved