OFFSET
0,2
COMMENTS
For n > 0, also numbers n such that A292032(n) = 1.
It is conjectured that A035105(n) is always a member of this sequence for n >= 4 but this remains unproved.
This is the complement of (1 + 2Z) U (2 + 3Z) U (3 + 5Z) U (5 + 8Z) U ..., see also the Example section. - M. F. Hasler, Feb 25 2018
LINKS
Ely Golden, Table of n, a(n) for n = 0..10000
FORMULA
a(10^7) = 45721410, a(10^8) = 457214230, a(10^9) = 4572142416. - Jacques Tramu, Feb 26 2018
EXAMPLE
a(2) = 6 since 6 mod 2 = 0, 6 mod 3 = 0, 6 mod 5 = 1, and 6 mod 8 = 6. (No other terms of A000045 need to be checked since the "illegal congruences" are all greater than 6, yet 6 is always congruent to 6 for those terms.)
From M. F. Hasler, Feb 26 2018: (Start)
This set can be constructed using a sieve which removes:
- first all numbers == 1 (mod 2), there remain the even numbers 0, 2, 4...;
- then all numbers == 2 (mod 3), i.e., == 2 (mod 6), there remain the numbers == 0 or 4 (mod 6): 0, 4, 6, 10, 12, 16, 18, 22, 24, 28, ...;
- then all numbers == 3 (mod 5), i.e., == 8 (mod 10), these are the numbers == 18 or 28 (mod 30), there remain numbers == 0, 4, 6, 10, 12, 16, 22 or 24 (mod 30);
- then all those == 5 (mod 8), but all these are odd;
- then all those == 8 (mod 13), i.e., == 8 (mod 26): there are 8 of these in [1..30*13], and there remain 8*(13-1) residue classes mod 30*13.
- then all those == 13 (mod 21): there are 48 of these left in [1..30*13*7], and there remain 8*12*7-48 = 48*(14-1) residue classes mod 30*13*7.
- then again there are none to remove == 21 (mod 34);
- then those == 34 (mod 55): these are 12*13 of the remaining 48*13*11 residue classes mod 30*13*7*11, so there remain 12*13*(4*11-1) of these; and so on.
This yields as upper bound of the asymptotic density: 1/2 * 2/3 * 4/5 * 12/13 * 13*14 * 43/44 ~ 0.223, the actual value is 0.2187...
(End)
MATHEMATICA
{0}~Join~Select[Range[3, 250], Function[n, NoneTrue[Block[{k = {1, 1}}, While[Last@ k <= n, AppendTo[k, Total@ Take[k, -2]]]; Partition[Most@ k, 2, 1]], Mod[n, #2] == #1 & @@ # &]]] (* Michael De Vlieger, Mar 19 2018 *)
PROG
(PARI) is_A292794(n, F=1)=!for(k=3, oo, F==n%(F=fibonacci(k))&&return; F>n&&break) \\ M. F. Hasler, Feb 25 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ely Golden, Sep 23 2017
STATUS
approved