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

%I A119242 #17 Jul 10 2020 22:05:10
%S A119242 1,2,5,31,234,1822,3611,17329,1511067,524827,180469424,472532614,
%T A119242 78102676912
%N A119242 Least number k such that there are exactly n powerful numbers between k^2 and (k+1)^2.
%C A119242 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.
%C A119242 a(10) is greater than 10^8. - _Giovanni Resta_, May 11 2006
%C A119242 a(n) > 10^11 for n >= 13. - _Donovan Johnson_, Sep 03 2013
%C A119242 Shiu (1980) proved that infinitely many values of k exist for every n. Therefore this sequence is infinite. - _Amiram Eldar_, Jul 10 2020
%D A119242 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.
%H A119242 Donovan Johnson, <a href="/A119242/a119242.txt">Powerful numbers between k^2 and (k+1)^2</a>
%H A119242 Steve Pettigrew, <a href="http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ55787.pdf">Sur la distribution de nombres speciaux consecutifs</a>, M.Sc. Thesis, Univ. Laval, 2000.
%H A119242 P. Shiu, <a href="https://doi.org/10.1112/S0025579300010056">On the number of square-full integers between successive squares</a>, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.
%e A119242 a(3) = 31 because 968, 972 and 1000 are between 961 and 1024.
%t A119242 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]}]
%Y A119242 Cf. A001694, A119241.
%K A119242 nonn,more
%O A119242 0,2
%A A119242 _T. D. Noe_, May 09 2006
%E A119242 a(8) and the previously known a(9) from _Giovanni Resta_, May 11 2006
%E A119242 a(10)-a(11) from _Donovan Johnson_, Dec 07 2008
%E A119242 a(12) from _Donovan Johnson_, Sep 01 2013