A060355 Numbers k such that k and k+1 are powerful numbers.

8, 288, 675, 9800, 12167, 235224, 332928, 465124, 1825200, 11309768, 384199200, 592192224, 4931691075, 5425069447, 13051463048, 221322261600, 443365544448, 865363202000, 8192480787000, 11968683934831, 13325427460800, 15061377048200, 28821995554247

Offset: 1

Author

Jason Earls, Apr 01 2001

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

Primitive elements are in A199801.

Cf. A001694, A060859.

Cf. A076446 (first differences of A001694).

Programs

Haskell

a060355 n = a060355_list !! (n-1)
a060355_list = map a001694 $ filter ((== 1) . a076446) [1..]
-- Reinhard Zumkeller, Jun 03 2015, Nov 30 2012

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 *)

PARI

is(n)=ispowerful(n)&&ispowerful(n+1) \\ Charles R Greathouse IV, Nov 16 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

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