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A060355 Numbers n such that n and n+1 are a pair of consecutive powerful numbers. 10
8, 288, 675, 9800, 12167, 235224, 332928, 465124, 1825200, 11309768, 384199200, 592192224, 4931691075, 5425069447, 13051463048, 221322261600, 443365544448, 865363202000, 8192480787000, 11968683934831, 13325427460800, 15061377048200, 28821995554247 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

"Erdős conjectured in 1975 that there do not exist three consecutive powerful integers." - Guy

See Guy for Erdős' conjecture and statement that this sequence is infinite. - Jud McCranie, Oct 13 2002

It is easy to see that this sequence is infinite: if n is in the sequence, so is 4*n*(n+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 in this sequence such that neither a(n) nor a(n+1) is a square. - Charles R Greathouse IV, Nov 19 2012

Paul Erdős wrote of meeting Kurt Mahler: "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

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

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.

J. J. O'Connor and E. F. Robertson, Biography of Kurt Mahler

C. Rivera, Problem 53.  Powerful numbers revisited, Prime Puzzles

Eric Weisstein's World of Mathematics, Powerful numbers

EXAMPLE

1825200 belongs to the sequence because 1825200 = 2.2.2.2.3.3.3.5.5.13.13, 1825201 = 7.7.193.193=1351^2 and both 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 *)

PROG

(PARI) is(n)=ispowerful(n)&&ispowerful(n+1) \\ Charles R Greathouse IV, Nov 16 2012

(Haskell)

import Data.List (elemIndices)

a060355 n = a060355_list !! (n-1)

a060355_list = map (a001694 . (+ 1)) $ elemIndices 1 a076446_list

-- Reinhard Zumkeller, 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

Primitive elements are in A199801.

Cf. A001694, A060859.

Cf. A076446 (first differences of A001694).

Sequence in context: A136364 A089670 A221612 * A060859 A187289 A187191

Adjacent sequences:  A060352 A060353 A060354 * A060356 A060357 A060358

KEYWORD

nonn

AUTHOR

Jason Earls (zevi_35711(AT)yahoo.com), 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

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Last modified February 27 18:57 EST 2015. Contains 255024 sequences.