

A119242


Least number k such that there are exactly n powerful numbers between k^2 and (k+1)^2.


8



1, 2, 5, 31, 234, 1822, 3611, 17329, 1511067, 524827, 180469424, 472532614, 78102676912
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OFFSET

0,2


COMMENTS

Pettigrew gives a(1)a(6) in table 14. He conjectures that k exists for every n. Surprisingly, a(8) is greater than 10^6, but a(9)=524827. The Mathematica program creates all powerful numbers <= nMax by computing all products of the form x^2 y^3.
a(10) is greater than 10^8.  Giovanni Resta, May 11 2006
a(n) > 10^11 for n >= 13.  Donovan Johnson, Sep 03 2013
Shiu (1980) proved that infinitely many values of k exist for every n. Therefore this sequence is infinite.  Amiram Eldar, Jul 10 2020


REFERENCES

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.


LINKS

Table of n, a(n) for n=0..12.
Donovan Johnson, Powerful numbers between k^2 and (k+1)^2
Steve Pettigrew, Sur la distribution de nombres speciaux consecutifs, M.Sc. Thesis, Univ. Laval, 2000.
P. Shiu, On the number of squarefull integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171178.


EXAMPLE

a(3) = 31 because 968, 972 and 1000 are between 961 and 1024.


MATHEMATICA

nMax=10^12; lst={}; Do[lst=Join[lst, i^3 Range[Sqrt[nMax/i^3]]^2], {i, nMax^(1/3)}]; lst=Union[lst]; n=0; k=1; Do[n0=k; While[lst[[k]]<j^2, k++ ]; n1=k; If[n1n01==n, Print[{n, j1}]; n++ ], {j, Sqrt[nMax]}]


CROSSREFS

Cf. A001694, A119241.
Sequence in context: A261750 A189559 A077483 * A068145 A032112 A058009
Adjacent sequences: A119239 A119240 A119241 * A119243 A119244 A119245


KEYWORD

nonn,more


AUTHOR

T. D. Noe, May 09 2006


EXTENSIONS

a(8) and the previously known a(9) from Giovanni Resta, May 11 2006
a(10)a(11) from Donovan Johnson, Dec 07 2008
a(12) from Donovan Johnson, Sep 01 2013


STATUS

approved



