

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
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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. AdamsWatters, 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 !! (n1)
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



