A240231 -id:A240231 - cp's OEIS frontend

A294933 Compound filter related to base-3 expansion of the exponents in prime factorization of n: a(n) = P(A294932(n), A294931(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 2, 2, 3, 2, 7, 2, 4, 3, 7, 2, 5, 2, 7, 7, 16, 2, 5, 2, 5, 7, 7, 2, 16, 3, 7, 4, 5, 2, 29, 2, 8, 7, 7, 7, 10, 2, 7, 7, 16, 2, 29, 2, 5, 5, 7, 2, 67, 3, 5, 7, 5, 2, 16, 7, 16, 7, 7, 2, 12, 2, 7, 5, 6, 7, 29, 2, 5, 7, 29, 2, 8, 2, 7, 5, 5, 7, 29, 2, 67, 16, 7, 2, 12, 7, 7, 7, 16, 2, 12, 7, 5, 7, 7, 7, 23, 2, 5, 5, 10, 2, 29, 2, 16, 29, 7, 2, 8, 2, 29, 7, 67

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

For all i, j: a(i) = a(j) => A038148(i) = A038148(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A294931, A294932.
Cf. also A293225, A293226 and A293442 (analogous filter for base-2).
Cf. A010057, A038148, A240231.

a(n) = (1/2)*(2 + ((A294932(n) + A294931(n))^2) - A294932(n) - 3*A294931(n)).

A240230 Table for the unique factorization of integers >= 2 into terms of A186285 or their squares.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, May 15 2014

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, ...].

			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)

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

Cf. A050376, A186285, A213925, A240231.

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 *)

cp's OEIS Frontend

A294933 Compound filter related to base-3 expansion of the exponents in prime factorization of n: a(n) = P(A294932(n), A294931(n)), where P(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula