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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

"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

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Last modified September 22 20:55 EDT 2014. Contains 247086 sequences.