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 A230030 Numbers n not divisible by 5 such that n^2 written in base 5 has no digit > 1. 5

%I

%S 1,972799,3051273374,6132750376,839228035909,3818357814376,

%T 2384643515634376,1490173338867234376,931329727148437734376,

%U 582077503203735352734376,363797992467864990240234376

%N Numbers n not divisible by 5 such that n^2 written in base 5 has no digit > 1.

%C If P(x) = 25x^4 + 15x^3 - 4x^2 + 3x + 1 then P(5^k) belongs to the sequence for every k > 2.

%C The initial condition is added to avoid trivial solutions of the form a(k)*5^m (m>0), whose square would always have the digits 1 and 0 in base 5. The previous subsequence of solutions P(5^k) consists in numbers written "10{k}24{k}10{k-1}30{k-1}1" in base 5, where "d{k}" means "digit d repeated k times". These terms (written in base 10) end in ...376. For k=8 this yields 582077503203735352734376 which might be the next term of the sequence. See A257283 and A257284 for the (less interesting) base 3 and base 4 analog. For the b=7 analog, the smallest nontrivial term is 20; for b=8 the first nontrivial terms are 3 and 11677. What are the subsequent terms, and the smallest nontrivial term for the b=6 analog? - _M. F. Hasler_, May 02 2015

%C Conjecture: a(k) = P(5^(k-2)) for every k > 5. - _David Radcliffe_, Sep 14 2018.

%H J. M. Borwein, Y. Bugeaud, and M. Coons, <a href="http://www.ams.org/notices/201505/rnoti-p526.pdf">The legacy of Kurt Mahler</a>, Notices of the American Mathematical Society, 62 5 (2015), 526-531.

%H Keith G. Calkins, <a href="/A230030/a230030_1.pdf">972799_10^2 = 111001100000110101_5</a>, Letter to the Editor, Notices Amer. Math. Soc., Vol. 62, No. 9 (2015), page 1029 (extract from full pdf).

%H D. Radcliffe, <a href="http://www.convex.org/wp/?p=18">Mahler's Quinary Conundrum</a>

%H David Radcliffe, <a href="/A230030/a230030.pdf">Mahler's Quinary Conundrum</a> [Cached copy]

%e 972799 belongs to the sequence because 972799^2 = 111001100000110101111001100000110101 (base 5).

%o (PARI) is(n)=n%5 && vecmax(digits(n^2,5))<2 \\ _Charles R Greathouse IV_, May 01 2015

%Y A262559 and A262560 are closely related.

%K nonn,base,more

%O 1,2

%A _David Radcliffe_, May 01 2015

%E a(10) from _David Radcliffe_, Dec 19 2015

%E a(11) from _David Radcliffe_, Sep 14 2018

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Last modified February 19 20:17 EST 2019. Contains 320328 sequences. (Running on oeis4.)