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

Author

T. D. Noe, May 09 2006

Keywords

• nonn
• more

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

Examples

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

References

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

Links

• 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.

Crossrefs

Cf. A001694, A119241.

Programs

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]]
				

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