A164841 Numbers whose squares have a digit average greater than 8.

3, 94863, 987917, 3162083, 29983327, 99477133, 99483667, 994927133, 2428989417, 2754991833, 2983284917, 2999833327, 3157196367, 9380293167, 9486778167, 28105157886, 31144643167, 31304790167, 31459487917, 31464263856, 94286790167, 99497231067, 244272388937

Offset: 1

Author

Zak Seidov, Aug 28 2009

Keywords

• base
• nonn

Comments

There are just 13 terms < 32*10^8.

3 is the only term whose square has a digit average that is an integer.

The squares of the first few terms have digit averages 9, 8.1, 8.08333, 8.15385, 8.06667, 8.125, 8.125, 8.22222, 8.05263, 8.05263, 8.10526, 8.10526, 8.05263, ...

The sequence contains all numbers of the form floor(30*100^k - 10^k*5/3), k > 5. As of today, we know of only 9 numbers whose square has a digit mean above 8.25: 3, 707106074079263583, 943345110232670883, 94180040294109027313, 2976388751488907738914, 312713447088224669275583, 893241282627485818275387, 314610537013606681884298837387 and 9984988582817657883693383344833. - M. F. Hasler, Apr 11 and Apr 13 2023

Links

• Ed Pegg, 314610537013606681884298837387, math-fun mailing list, April 11, 2023.

Crossrefs

Cf. A164771 (s=1), A164770 (s=2), A164782 (s=3), A164776 (s=4), A164774 (s=5), A164779 (s=6), A164773 (s=7), A164772 (s=8).

Cf. A164842 (s < 1).

Programs

PARI

for(L=1,oo, for(n=sqrtint(10^(L-1)-1)+1,sqrtint(10^L-1), sumdigits(n^2) > 8*L && print1(n", "))) \\ M. F. Hasler, Apr 11 2023

Extensions

a(14)-a(23) from Lars Blomberg, Apr 29 2013