A300005
Index of first occurrence of 2n in A300004 (or -1 of 2n does not occur), where A300004 are first differences of numbers not congruent to A000045(k) modulo A000045(k+1) for any k > 1.
Original entry on oeis.org
1, 0, 5, 26, 14, 100, 323, 1671, 4293, 10934, 208741, 753123, 627460, 87918559, 1137656208
Offset: 1
Index n| gap 2n | a(n) | A292794(a(n)) with a(n) = the smallest k
1 | 2 | 1 | 4 such that A300004(k) = 2n
2 | 4 | 0 | 0 = A292794(k+1) - A292794(k)
3 | 6 | 5 | 16
4 | 8 | 26 | 106
5 | 10 | 14 | 54
6 | 12 | 100 | 444
7 | 14 | 323 | 1456
8 | 16 | 1671 | 7614
9 | 18 | 4293 | 19602
10 | 20 | 10934 | 49966
11 | 22 | 208741 | 954384
12 | 24 | 753123 | 3443356
13 | 26 | 627460 | 2868820
14 | 28 | 87918559 | 401976096
15 | 30 |1137656208| 5201526136
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A300005=List();b=c=L=0;ng=2;for(n=1,oo, is_A292794(n)||next; c++; bittest(b,g=-L+L=n)&&next; b+=2^g; listput(A300005,[g,c-1,n-g]); g>ng&&next; listsort(A300005); for(i=ng/2,#A300005, A300005[i][1]>ng&&break; printf("%d,",A300005[i]);ng+=2))
A292794
Numbers not congruent to A000045(k) mod A000045(k+1) for all k > 1.
Original entry on oeis.org
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
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)
Cf.
A300004 for the sequence of first differences.
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{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 *)
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is_A292794(n,F=1)=!for(k=3,oo,F==n%(F=fibonacci(k))&&return;F>n&&break) \\ M. F. Hasler, Feb 25 2018
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