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

1, 972799, 3051273374, 6132750376, 839228035909, 3818357814376, 2384643515634376, 1490173338867234376, 931329727148437734376, 582077503203735352734376, 363797992467864990240234376

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

Table of n, a(n) for n=1..11.

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

A262559 and A262560 are closely related.

Sequence in context: A121888 A178292 A237337 * A206116 A230538 A254089

Adjacent sequences: A230027 A230028 A230029 * A230031 A230032 A230033

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