A333337 Indices of rows of n consecutive smallest primes in A333238, or -1 if n consecutive smallest primes do not appear in A333238.

0, 1, 2, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 25, 27, 24, 28, 33, 35, 30, 39, 44, 45, 49, 51, 55, 63, 57, 65, 69, 75, 77, 81, 85, 60, 76, 87, 91, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 135, 143, 145, 147, 153, 155, 161, 169, 159, 165, 171, 175, 177

Offset: 0

Author

Michael De Vlieger, David James Sycamore, May 25 2020

Keywords

• tabf
• sign

Comments

Consider the irregular table where row m lists the distinct smallest primes p of prime partitions of m. Row n of this sequence contains all m that have n consecutive primes starting with 2.

Alternatively, positions of k-repunits in A333259.

A330507(n) = First terms in row n.

Null rows occur at n = {90, 151, 349, 352, 444, ...} and are thus filled with the term -1.

Examples

			Table begins:
       0:    0     1
       1:    2     4
       2:    6     8     9
       3:   10    12    15    16
       4:   18    20    21    25    27
       5:   24    28    33    35
       6:   30    39    44    45    49
       7:   51    55    63
       8:   57    65
       9:   60    76    87    91    95
      10:   69    75    77    81    85
      11:   99   105
      12:  111   115   117   119   121
      13:  123   125   135
      14:  143   145
      15:  147   153   155   161   169
      16:  159   165   171   175
      17:  177   185   187
Consider the table plotting prime p in row m of A333238 at pi(p) place; intervening primes missing from row m are shown by "." as a place holder:
    m      Primes in row m of A333238
    ---------------------------------
    2:     2
    3:     .   3
    4:     2
    5:     2   .   5
    6:     2   3
    7:     2   .   .   7
    8:     2   3
    9:     2   3
    10:    2   3   5
    11:    2   3   .   .  11
    12:    2   3   5
    13:    2   3   .   .   .  13
    14:    2   3   .   7
    15:    2   3   5
    16:    2   3   5
    17:    2   3   5   .   .   .  17
    ...
There are no primes in rows 0 or 1 of A333238, thus row 0 of this sequence contains {0, 1}.
The smallest prime, 2, appears alone in rows 2 and 4 of A333238, thus row 1 of this sequence contains {2, 4}.
We have the primes {2, 3} and no other primes in rows {6, 8, 9} in A333238, thus row 2 of this sequence contains {6, 8, 9}.
We have the primes {2, 3, 5} and no other primes in rows {10, 12, 15, 16} in A333238, thus row 3 of this sequence contains {10, 12, 15, 16}, etc.
		

Crossrefs

Cf. A330507, A333238, A333259.

Programs

Mathematica

Block[{m = 120, s, a}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], Last@ s], a = ReplacePart[a, # -> Union@ Append[a[[#]], Last@ s]], Nothing]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]]] &@ Total[s], {i, Infinity}]; s = {0}~Join~Map[Which[Length@ # == 0, 0, And[Length@ # == 1, First@ # == 2], 1, True, If[Union@ # == {1}, Length@ # + 1, -1] &[Differences@ PrimePi@ #, {} -> {2}]] &, a]; Array[-1 + Position[s, #][[All, 1]] /. k_ /; MissingQ@ k -> {-1} &, Max@ s + 1, 0]]