A333337 - OEIS

%I #10 Aug 29 2021 01:56:33

%S 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,

%T 51,55,63,57,65,69,75,77,81,85,60,76,87,91,95,99,105,111,115,117,119,

%U 121,123,125,135,143,145,147,153,155,161,169,159,165,171,175,177

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

%C 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.

%C Alternatively, positions of k-repunits in A333259.

%C A330507(n) = First terms in row n.

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

%e Table begins:

%e        0:    0     1

%e        1:    2     4

%e        2:    6     8     9

%e        3:   10    12    15    16

%e        4:   18    20    21    25    27

%e        5:   24    28    33    35

%e        6:   30    39    44    45    49

%e        7:   51    55    63

%e        8:   57    65

%e        9:   60    76    87    91    95

%e       10:   69    75    77    81    85

%e       11:   99   105

%e       12:  111   115   117   119   121

%e       13:  123   125   135

%e       14:  143   145

%e       15:  147   153   155   161   169

%e       16:  159   165   171   175

%e       17:  177   185   187

%e 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:

%e     m      Primes in row m of A333238

%e     ---------------------------------

%e     2:     2

%e     3:     .   3

%e     4:     2

%e     5:     2   .   5

%e     6:     2   3

%e     7:     2   .   .   7

%e     8:     2   3

%e     9:     2   3

%e     10:    2   3   5

%e     11:    2   3   .   .  11

%e     12:    2   3   5

%e     13:    2   3   .   .   .  13

%e     14:    2   3   .   7

%e     15:    2   3   5

%e     16:    2   3   5

%e     17:    2   3   5   .   .   .  17

%e     ...

%e There are no primes in rows 0 or 1 of A333238, thus row 0 of this sequence contains {0, 1}.

%e The smallest prime, 2, appears alone in rows 2 and 4 of A333238, thus row 1 of this sequence contains {2, 4}.

%e 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}.

%e 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.

%t 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]]

%Y Cf. A330507, A333238, A333259.

%K tabf,sign

%O 0,3

%A _Michael De Vlieger_, _David James Sycamore_, May 25 2020