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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 (list; graph; refs; listen; history; text; internal format)
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.

Alexandra Hercilia Pereira Silva, in reply to Z. 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

Cf. A001043, A016789, A016885, A017041, A017473, A269100.

Sequence in context: A031445 A131547 A044085 * A107291 A044466 A022274

Adjacent sequences:  A319521 A319522 A319523 * A319525 A319526 A319527

KEYWORD

nonn,look

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

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Last modified December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)