A292794 Numbers not congruent to A000045(k) mod A000045(k+1) for all k > 1.
0, 4, 6, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 64, 66, 70, 72, 82, 84, 90, 94, 96, 100, 102, 106, 114, 120, 124, 126, 130, 132, 136, 142, 150, 154, 156, 162, 166, 172, 174, 180, 184, 186, 192, 196, 204, 210, 214, 220, 222, 226, 232, 234, 240, 246, 250, 252, 256
Offset: 0
Examples
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)
Links
- Ely Golden, Table of n, a(n) for n = 0..10000
- Ely Golden, Python program for generating terms of this sequence
Programs
-
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 *) -
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
Formula
a(10^7) = 45721410, a(10^8) = 457214230, a(10^9) = 4572142416. - Jacques Tramu, Feb 26 2018
Comments