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A319524
a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + m*prime(n+1) and prime(n+1) + m*prime(n+2), m >= 1, n >= 1.
2
8, 33, 40, 128, 115, 302, 226, 226, 835, 401, 734, 1718, 1030, 842, 3121, 3475, 1401, 2339, 5108, 1969, 3233, 2486, 6491, 9692, 10298, 5560, 11552, 6211, 4177, 7987, 6022, 18763, 16678, 21893, 8001, 25585, 13523, 9682, 30961, 32035, 7057, 36089, 19105, 39002, 7162, 47041, 50163, 51752
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 GĂ©rard, 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
nonn,look
AUTHOR
EXTENSIONS
Table from Jon E. Schoenfield, Sep 23 2018
More terms from Amiram Eldar, Sep 22 2018
STATUS
approved