OEIS A333337

Positions 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

0, 2.

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
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.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 115.

OEIS Wiki, Full reptend primes.
Eric Weisstein's World of Mathematics, Cyclic Number.
Eric Weisstein's World of Mathematics, Full Reptend Prime.

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, interposing 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.

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]]

Cf. A330507, A333238, A333259.

nonn, tabf