|
| |
|
|
A240230
|
|
Table for the unique factorization of integers >= 2 into terms of A186285 or their squares.
|
|
2
|
|
|
|
1, 2, 3, 2, 2, 5, 2, 3, 7, 8, 3, 3, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 8, 17, 2, 3, 3, 19, 2, 2, 5, 3, 7, 2, 11, 23, 3, 8, 5, 5, 2, 13, 27, 2, 2, 7, 29, 2, 3, 5, 31, 2, 2, 8, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 37, 2, 19, 3, 13, 5, 8, 41, 2, 3, 7, 43, 2, 2, 11, 3, 3, 5, 2, 23, 47, 2, 3, 8, 7, 7, 2, 5, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
The terms of A186285 are primes to powers of 3 (PtPP(p=3) primes to prime powers with p=3). See A050376 for PtPP(2), appearing in the OEIS as 'Fermi-Dirac' primes, because in this case the unique representation of n >= 2 works with distinct members of A050376, hence the multiplicity (occupation number) is either 0 (not present) or 1 (appearing once). For p=3 the multiplicities are 0, 1, 2. See the multiplicity sequences given in the examples. At position m the multiplicity for A186285(m), m >= 1, is recorded, and trailing zeros are omitted, except for n = 1.
In order to include n=1 one defines as its representation 1, even though 1 is not a member of A186285 (in order to have a unique representation for n >= 2 modulo commutation of factors).
The length of row n, the number of factors) is obtained from the (reversed) base 3 representation of the exponents of the primes appearing in the ordinary factorization of n, by adding all entries. E.g., n = 2^5*5^7 = 2500000 will have row length 6 because (5)_(3r) = [2, 1] and (7)_(3r) = [1, 2] (reversed base 3), leading to the 6 factors (2^2*8^1)*(5^1*125^2) = 2*2*5*8*125*125. The row length sequence is A240231 = [1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, ...].
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
The irregular triangle a(n,k) starts (in the first part the factors are listed):
n\k 1 2 3 ... multiplicity sequence
1: 1 0-sequence [repeat(0,)]
2: 2 [1]
3: 3 [0, 1]
4: 2, 2 [2]
5: 5 [0, 0, 1]
6: 2, 3 [1, 1]
7: 7 [0, 0, 0, 1]
8: 8 [0, 0, 0, 0, 1]
9: 3, 3 [0, 2]
10: 2, 5 [1, 0, 1]
11: 11 [0, 0, 0, 0, 0, 1]
12: 2, 2, 3 [2, 1]
13: 13 [0, 0, 0, 0, 0, 0, 1]
14: 2, 7 [1, 0, 0, 1]
15: 3, 5 [0, 1, 1]
16: 2, 8 [1, 0, 0, 0, 1]
17: 17 [0, 0, 0, 0, 0, 0, 0, 1]
18: 2, 3, 3 [1, 2]
19: 19 [0, 0, 0, 0, 0, 0, 0, 0, 1]
20: 2, 2, 5 [2, 0, 1]
|
|
|
MATHEMATICA
|
With[{s = Select[Select[Range[53], PrimePowerQ], IntegerQ@Log[3, FactorInteger[#][[1, -1]]] &]}, {{1}}~Join~Table[Reverse@ Rest@ NestWhileList[Function[{k, m}, {k/#, #} &@ SelectFirst[Reverse@ TakeWhile[s, # <= k &], Divisible[k, #] &]] @@ # &, {n, 1}, First@ # > 1 &][[All, -1]], {n, 2, Max@ s}]] // Flatten (* Michael De Vlieger, Aug 14 2017 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
