A331788 a(n) is the smallest m such that for any N, at least one of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N.

1, 3, 3, 7, 9, 9, 13, 15, 9, 19, 39, 39, 79, 99, 99, 139, 159, 99, 199, 399, 399, 799, 999, 999, 1399, 1599, 999, 1999, 3999, 3999, 7999, 9999, 9999, 13999, 15999, 9999, 19999, 39999, 39999, 79999, 99999, 99999, 139999, 159999, 99999, 199999, 399999, 399999, 799999

Offset

1,2

Comments

The main sequence is A331786; this is added because some people may search for this.

Links

Jianing Song, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Formula

If n = 9*s + t, 1 <= t <= 9, then a(n) = 10^s*(2*t-gcd(t,9)+1) - 1. See A331787 for a proof of the formula in base b.

a(n) = A331786(n) + 1.

Conjectures from Colin Barker, Jan 26 2020: (Start)

G.f.: x*(1 + 2*x + 4*x^3 + 2*x^4 + 4*x^6 + 2*x^7 - 6*x^8) / ((1 - x)*(1 - 10*x^9)).

a(n) = a(n-1) + 10*a(n-9) - 10*a(n-10) for n>10.

(End) [This conjecture is correct.]

a(n) = O(10^(n/9)).

Example

See A331786.

Prog

(PARI) a(n) = my(s=(n-1)\9, t=(n-1)%9+1); 10^s*(2*t-gcd(t, 9)+1)-1

Crossrefs

Cf. A007953 (S(N)), A051885, A331786.

Row 10 of A331789.

Sequence in context: A143088 A360875 A131036 * A206433 A301589 A237666

Adjacent sequences: A331785 A331786 A331787 * A331789 A331790 A331791

Keyword

nonn,base,easy

Author

Jianing Song, Jan 25 2020

Status

approved