A119242 Least number k such that there are exactly n powerful numbers between k^2 and (k+1)^2.

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

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 square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

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[n1-n0-1==n, Print[{n, j-1}]; 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