|
|
A333337
|
|
Indices of rows of n consecutive smallest primes in A333238, or -1 if n consecutive smallest primes do not appear in A333238.
|
|
0
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
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.
Null rows occur at n = {90, 151, 349, 352, 444, ...} and are thus filled with the term -1.
|
|
LINKS
|
|
|
EXAMPLE
|
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:
---------------------------------
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.
|
|
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]]
|
|
CROSSREFS
|
|
|
KEYWORD
|
tabf,sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|