OFFSET
1,1
COMMENTS
Construct a table T in which T(m,n) = prime(n) + m*prime(n+1) as shown below. Then a(n) is defined as the smallest number appearing both in column n and column n+1, so a(1)=8, a(2)=33, a(3)=40, etc.
.
m\n| 1 2 3 4 5 6 7 8 ...
----+--------------------------------------------------
1 | 5 --8 12 18 24 30 36 42 ...
|
2 | 8-- 13 19 29 37 47 55 65 ...
|
3 | 11 18 26 40 50 64 74 88 ...
| /
4 | 14 23 33 / 51 63 81 93 111 ...
| / /
5 | 17 28 / 40- 62 76 98 112 134 ...
| /
6 | 20 33- 47 73 89 115 131 157 ...
| /
7 | 23 38 54 84 102 / 132 150 180 ...
| /
8 | 26 43 61 95 115 149 169 203 ...
|
9 | 29 48 68 106 128 166 188 226 ...
| / /
10 | 32 53 75 117 / 141 183 207 / 249 ...
| / /
11 | 35 58 82 128 154 200 226 272 ...
|
12 | 38 63 89 139 167 217 245 295 ...
|
13 | 41 68 96 150 180 234 264 318 ...
|
14 | 44 73 103 161 193 251 283 341 ...
|
15 | 47 78 110 172 206 268 302 364 ...
| /
16 | 50 83 117 183 219 285 / 321 387 ...
| /
17 | 53 88 124 194 232 302 340 410 ...
|
... |... ... ... ... ... ... ... ... ...
Conjectures:
1. There are infinitely many pairs of consecutive equal terms. (Note that the first pair is (a(7), a(8)).)
2. There exists no N such that the sequence is monotonic for n > N.
From Amiram Eldar, Sep 22 2018: (Start)
Theorem 1: The intersection of the two mentioned arithmetic progressions is always nonempty.
Corollary: The sequence is infinite. (End)
Sequences that derive from this:
1. Positions in {s(n)} at which a(n) occurs: (2,6,5,11,8,17,19,...).
2. Positions in {s(n+1)} at which a(n) occurs: (1,4,3,9,6,15,15,...).
3. Differences between these two sequences: (1,2,2,2,2,4,...).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 600 terms from Muniru A Asiru)
Fourth International contest of logical problems, Problem 7, the Ludomind Society.
Fifth International contest of logical problems, Problem 6, the Ludomind Society, 2009.
Olivier Gérard, in reply to Zak Seidov, 11 related sequences, SeqFan list, Apr 14 2016.
MATHEMATICA
a[n_]:=ChineseRemainder[{Prime[n], Prime[n+1]}, {Prime[n+1], Prime[n+2]} ]; Array[a, 44] (* Amiram Eldar, Sep 22 2018 *)
PROG
(GAP) P:=Filtered([1..10000], IsPrime);;
T:=List([1..Length(P)-1], n->List([1..Length(P)-1], m->P[n]+m*P[n+1]));;
a:=List([1..50], k->Minimum(List([1..Length(T)-1], i->Intersection(T[i], T[i+1]))[k])); # Muniru A Asiru, Sep 26 2018
CROSSREFS
KEYWORD
AUTHOR
Alexandra Hercilia Pereira Silva, Sep 22 2018
EXTENSIONS
Table from Jon E. Schoenfield, Sep 23 2018
More terms from Amiram Eldar, Sep 22 2018
STATUS
approved