A244859 Least positive multiple of n which when written in base 10 is either a repunit or of the form 111...000.

0, 1, 10, 111, 100, 10, 1110, 111111, 1000, 111111111, 10, 11, 11100, 111111, 1111110, 1110, 10000, 1111111111111111, 1111111110, 111111111111111111, 100, 111111, 110, 1111111111111111111111, 111000, 100, 1111110, 111111111111111111111111111, 11111100

Offset

0,3

Comments

a(1017) has 1008 digits. - Michael S. Branicky, Feb 22 2024

a(0) = 0 by convention: It can be considered as a repunit with zero digits, A002275(0) = (10^0-1)/9, and it is a positive multiple of n in the sense of k*n with k > 0. - M. F. Hasler, Mar 04 2025

Links

Michael S. Branicky, Table of n, a(n) for n = 0..1016 (terms 101..936 from Robert Israel, terms 0..100 from Chai Wah Wu).

Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.

Formula

a(n) = n*A244927(n). - M. F. Hasler, Mar 04 2025

a(3^k) = (10^(3^k)-1)/9. For n > 0, A055642(a(n)) <= n. If n > 2 is not a power of 3, then A055642(a(n)) < n. - Chai Wah Wu, Mar 04 2025

Maple

A244859:= proc(n) local m, d2, d5;
d2:= padic:-ordp(n, 2);
d5:= padic:-ordp(n, 5);
m:= n/2^d2/5^d5;
10^max(d2, d5)*(10^numtheory:-order(10, 9*m)-1)/9
end proc:
A244859(0):= 0:
seq(A244859(n), n= 0..100); # Robert Israel, Jul 08 2014

Prog

(Python)
def a(n):
    if n == 0: return 0
    moddict = {0: 0}
    for e in range(1, n+2):
        repe = (10**e-1)//9
        r = repe%n
        if r in moddict:
            return repe - moddict[r]
        else:
            moddict[r] = repe
print([a(n) for n in range(29)]) # Michael S. Branicky, Feb 22 2024
(PARI) apply( {A244859(n, m=Map(Mat([0, 0])))=for(L=1, n, my(r=10^L\9); iferr(return(r-mapget(m, r%n)), E, mapput(m, r%n, r)))}, [0..33]) \\ M. F. Hasler, Mar 04 2025

Crossrefs

Equal to A004290 for n = 1 .. 6.

Cf. A002275, A244927.

Sequence in context: A228006 A004290 A257344 * A159551 A105991 A284352

Adjacent sequences: A244856 A244857 A244858 * A244860 A244861 A244862

Keyword

nonn,base,changed

Author

Chai Wah Wu, Jul 07 2014

Status

approved