A320122 Numbers that are not Keith numbers in any base.

12, 30, 390, 1170, 1200, 1560, 2340, 2760, 3120, 3900, 4680, 6120, 6240, 7680, 7800, 8460, 10020, 10140, 10950, 11580, 15090, 15480, 17160, 17580, 18360, 19140, 20280, 20700, 20940, 21480, 23040, 23280, 24060, 24210, 24960, 26550, 28740, 29250, 29520, 29670, 30060, 31080, 32400

Offset

1,1

Comments

A number N >= 2 is a Keith number in a base b <= N if the Fibonacci sequence u(i) whose initial terms are the t digits of N in the base b, and later terms are given by rule that u(i) = sum of t previous terms, contains N itself. Here a(n) is the n-th number N that is not a Keith number in any base b <= N.

Links

Table of n, a(n) for n=1..43.

Arie Bos, Inventory of n-step Fibonacci sequences, 2015.

Example

a(1) = 12 because 12 is not a Keith number in any base from 2 to 12, while all previous numbers are in some base.
For example, with b = 2, the sequence is : 1, 1, 0, 0, 2, 3, 5, 10, 20, ...; it doesn't contain 12. See A251703.

Maple

fibo:=proc(n, b) local L, m, M, k:
L:=convert(n, base, b):m:=nops(L):M:=seq(L[m+1-k], k=1..m):
while M[m]<n do M:=M, add(M[q], q=1..m): M:=seq(M[q], q=2..m+1) od:
if M[m]=n then true else false fi end:
test:=proc(n) local b:
for b from 2 to n do if fibo(n, b) then return(true) fi od:
return(false) end:
L:=NULL:for n from 2 to 1200 do if not(test(n)) then  L:=L, n fi od:L;

Prog

(Python)
def digits(n, b):
    r = []
    m = n
    while m > 0:
        r = [m % b] + r
        m = m // b
    return r
def fibo(n, b):
    L = digits(n, b)
    m = len(L) - 1
    while L[m] < n:
        L.append(sum(k for k in L))
        L.pop(0)
    return L[m] == n
def test(n):
    for b in range(2, n + 1):
        if fibo(n, b):
            return True
    return False
print([n for n in range(2, 2001) if not test(n)])
(PARI) iskb(n, b) = if(n<b, return(0)); my(v=digits(n, b), t=#v); while(v[#v]<n, v=concat(v, sum(i=0, t-1, v[#v-i]))); v[#v]==n; \\ after A007629
isok(n) = if (n<=2, 0, for(b=2, n-1, if (iskb(n, b), return(0))); return (1)); \\ Michel Marcus, Oct 08 2018

Crossrefs

Cf. A007629 (Keith numbers in base 10).

Sequence in context: A221520 A214311 A005147 * A007308 A065138 A279821

Adjacent sequences: A320119 A320120 A320121 * A320123 A320124 A320125

Keyword

nonn,base

Author

Robert FERREOL, Oct 06 2018

Extensions

More terms from Michel Marcus, Oct 08 2018

Status

approved