A333238 revision #34
|
|
A333238
|
|
Irregular table where row n lists the distinct smallest primes p of prime partitions of n.
|
|
9
|
|
|
2, 3, 2, 2, 5, 2, 3, 2, 7, 2, 3, 2, 3, 2, 3, 5, 2, 3, 11, 2, 3, 5, 2, 3, 13, 2, 3, 7, 2, 3, 5, 2, 3, 5, 2, 3, 5, 17, 2, 3, 5, 7, 2, 3, 5, 19, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 11, 2, 3, 5, 23, 2, 3, 5, 7, 11, 2, 3, 5, 7, 2, 3, 5, 7, 13, 2, 3, 5, 7, 2, 3, 5, 7, 11
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
COMMENTS
|
A prime partition of n is an integer partition wherein all parts are prime. For instance, (3 + 2) is a prime partition of the sum 5; for n = 5, (5) is also a prime partition. For 6, we have two prime partitions (3 + 3) and (2 + 2 + 2).
We note that there are no prime partitions for n = 1, therefore the offset of this sequence is 2.
The number of prime partitions of n is shown by A000607(n).
For prime p, row p includes p itself as the largest term, since p is the sum of (p).
In the irregular table below, T(n,k) is either prime(k) or is empty. The former means there is at least one prime partition of n with least part prime(k), the latter means that no such partition exists. T(n,k) empty is not recorded in the data.
Recursion for n >= 4: T(n,k) = prime(k) iff T((n-prime(k)), k) = prime(k), or there is a q > k such that T((n-prime(k)), q)) = prime(q); else T(n,k) is empty. Example: T(17,3) = 5 because T(12,3) = prime(3) = 5. T(10,2) = 3 since although T(7,2) is empty, T(7,4) = prime(4) = 7. (End)
|
|
LINKS
|
|
|
EXAMPLE
|
The least primes among the prime partitions of 5 are 2 and 5, cf. the 2 prime partitions of 5: (5) and (3, 2), thus row 5 lists {2, 5}.
The least primes among the prime partitions of 6 are 2 and 3, cf. the two prime partitions of 6, (3, 3), and (2, 2, 2), thus row 6 lists {2, 3}.
Row 7 contains {2, 7} because there are 3 prime partitions of 7: (7), (5, 2), (3, 2, 2). Note that 2 is the smallest part of the latter two partitions, thus only 2 and 7 are distinct.
Table plotting prime p in row n at pi(p) place, intervening primes missing from row n are shown by "." as a place holder:
n Primes in row n
----------------------
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
...
|
|
MAPLE
|
b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
end:
T:= proc(n) option remember; (p-> seq(`if`(isprime(i) and
coeff(p, x, i)>0, i, [][]), i=2..degree(p)))(b(n, 2, x))
end:
|
|
MATHEMATICA
|
Block[{a, m = 20, s}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], First@ s], a = ReplacePart[a, # -> 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}]; Union /@ a // Flatten]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
editing
|
|
|
|