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

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

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

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

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.

See Guy for Erdos' 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). [From 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 primepuzzles 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

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

primepuzzles, Problem 53.  Powerful numbers revisited

Eric Weisstein's World of Mathematics, Powerful numbers

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

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

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 November 26 08:46 EST 2014. Contains 250021 sequences.