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A240230 Table for the unique factorization of integers >= 2 into members 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 members 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 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

Michael De Vlieger, Table of n, a(n) for n = 1..13622 (Rows 1 <= n <= 5000).

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]

...(reformatted - Wolfdieter Lang, May 16 2014)

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

Cf. A050376, A186285, A213925, A240231.

Sequence in context: A225243 A207338 A027746 * A238689 A166454 A283239

Adjacent sequences:  A240227 A240228 A240229 * A240231 A240232 A240233

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, May 15 2014

STATUS

approved

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Last modified December 17 14:12 EST 2018. Contains 318201 sequences. (Running on oeis4.)