OFFSET
1,2
COMMENTS
If P(x) = 25x^4 + 15x^3 - 4x^2 + 3x + 1 then P(5^k) belongs to the sequence for every k > 2.
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 of 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
Conjecture: a(k) = P(5^(k-2)) for every k > 5. - David Radcliffe, Sep 14 2018
LINKS
J. M. Borwein, Y. Bugeaud, and M. Coons, The legacy of Kurt Mahler, Notices of the American Mathematical Society, 62 5 (2015), 526-531.
Keith G. Calkins, 972799_10^2 = 111001100000110101_5, Letter to the Editor, Notices Amer. Math. Soc., Vol. 62, No. 9 (2015), page 1029 (extract from full pdf).
David Radcliffe, Mahler's Quinary Conundrum
EXAMPLE
972799 belongs to the sequence because 972799^2 = 111001100000110101111001100000110101 (base 5).
PROG
(PARI) is(n)=n%5 && vecmax(digits(n^2, 5))<2 \\ Charles R Greathouse IV, May 01 2015
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
David Radcliffe, May 01 2015
EXTENSIONS
a(10) from David Radcliffe, Dec 19 2015
a(11) from David Radcliffe, Sep 14 2018
STATUS
approved