A186285 -id:A186285 - cp's OEIS frontend

A186287 a(n) is the denominator of the rational number whose "factorization" into terms of A186285 has the balanced ternary representation corresponding to n.

1, 1, 2, 1, 1, 6, 3, 3, 2, 1, 1, 2, 1, 1, 30, 15, 15, 10, 5, 5, 10, 5, 5, 6, 3, 3, 2, 1, 1, 2, 1, 1, 6, 3, 3, 2, 1, 1, 2, 1, 1, 105, 105, 105, 35, 35, 35, 35, 35, 35, 21, 21, 21, 7, 7, 7, 7, 7, 7, 21, 21, 21, 7, 7, 7, 7, 7, 7, 15, 15, 15, 5, 5, 5, 5, 5, 5, 3, 3, 3, 1, 1, 1, 1, 1, 1, 3, 3, 3

Offset: 0

Daniel Forgues, Feb 17 2011

Denominators from the ordering of positive rational numbers by increasing balanced ternary representation of the "factorization" of positive rational numbers into terms of A186285 (prime powers with a power of three as exponent).

			The balanced ternary digits {-1,0,+1} are represented here as {2,0,1}.
   n BalTern A186286/A186287 (in reduced form)
   0      0  Empty product = 1 = 1/1, a(n) = 1
   1      1  2 = 2/1,                 a(n) = 1
   2     12  3*(1/2) = 3/2,           a(n) = 2
   3     10  3 = 3/1,                 a(n) = 1
   4     11  3*2 = 6 = 6/1,           a(n) = 1
   5    122  5*(1/3)*(1/2) = 5/6,     a(n) = 6
   6    120  5*(1/3) = 5/3,           a(n) = 3
   7    121  5*(1/3)*2 = 10/3,        a(n) = 3
...    ...
  41  12222  8*(1/7)*(1/5)*(1/3)*(1/2) = 8/210 = 4/105, a(n) = 105

Daniel Forgues, Table of n, a(n) for n = 0..9841
OEIS Wiki, Orderings of rational numbers

Cf. A186285, A186286, A185169, A052330.

The balanced ternary representation of n
    n = Sum(i=0..1+floor(log_3(2|n|)) n_i * 3^i, n_i in {-1,0,1},
  is taken as the representation of the "factorization" of the positive rational number c(n)/d(n) into terms from A186285
    c(n)/d(n) = Prod(i=0..1+floor(log_3(2|n|)) (A186285(i+1))^(n_i), where A186285(i+1) is the (i+1)th prime power with exponent being a power of 3. Then a(n) is the denominator, i.e., d(n).

A186286 a(n) is the numerator of the rational number whose "factorization" into terms of A186285 has the balanced ternary representation corresponding to n.

Original entry on oeis.org

1, 2, 3, 3, 6, 5, 5, 10, 5, 5, 10, 15, 15, 30, 7, 7, 14, 7, 7, 14, 21, 21, 42, 7, 7, 14, 7, 7, 14, 21, 21, 42, 35, 35, 70, 35, 35, 70, 105, 105, 210, 4, 8, 16, 4, 8, 16, 12, 24, 48, 4, 8, 16, 4, 8, 16, 12, 24, 48, 20, 40, 80, 20, 40, 80, 60, 120, 240, 4, 8, 16, 4, 8, 16, 12, 24, 48, 4, 8

Offset: 0

Daniel Forgues, Feb 17 2011

Numerators from the ordering of positive rational numbers by increasing balanced ternary representation of the "factorization" of positive rational numbers into terms of A186285 (prime powers with a power of three as exponent).

			The balanced ternary digits {-1,0,+1} are represented here as {2,0,1}.
   n BalTern A186286/A186287 (in reduced form)
   0      0  Empty product = 1 = 1/1, a(n) = 1
   1      1  2 = 2/1,                 a(n) = 2
   2     12  3*(1/2) = 3/2,           a(n) = 3
   3     10  3 = 3/1,                 a(n) = 3
   4     11  3*2 = 6 = 6/1,           a(n) = 6
   5    122  5*(1/3)*(1/2) = 5/6,     a(n) = 5
   6    120  5*(1/3) = 5/3,           a(n) = 5
   7    121  5*(1/3)*2 = 10/3,        a(n) = 10
  ...   ...
  41  12222  8*(1/7)*(1/5)*(1/3)*(1/2) = 8/210 = 4/105, a(n) = 4

Daniel Forgues, Table of n, a(n) for n = 0..9841
OEIS Wiki, Orderings of rational numbers

Cf. A186285, A186287, A185169, A052330.

The balanced ternary representation of n
    n = Sum(i=0..1+floor(log_3(2|n|)) n_i * 3^i, n_i in {-1,0,1},
  is taken as the representation of the "factorization" of the positive rational number c(n)/d(n) into terms from A186285
    c(n)/d(n) = Prod(i=0..1+floor(log_3(2|n|)) (A186285(i+1))^(n_i), where A186285(i+1) is the (i+1)th prime power with exponent being a power of 3. Then a(n) is the numerator, i.e., c(n).

A185169 Cantor's ordering of positive rational numbers, where a(n) is the balanced ternary representation of the "factorization" of the positive rational number into terms of A186285.

Original entry on oeis.org

0, 2, 1, 20, 10, 20001, 21, 12, 10002, 200, 100, 22, 201, 20011, 10022, 102, 11, 2000, 210, 120, 1000, 20000, 2001, 10202, 20101, 1002, 10000, 20000000010, 2010, 1020, 10000000020, 202, 20000000011, 20010, 12002, 122, 211, 21001, 10020, 10000000022, 101, 200000, 2100, 1200, 100000, 20021, 200001, 212, 20000010012, 20100, 2011, 1022, 10200, 10000020021, 121, 100002, 10012

Offset: 1

Daniel Forgues, Feb 19 2011

nonn

The balanced ternary digits {-1,0,+1} are represented here as {2,0,1}.

The "factorization" of positive rational numbers into prime powers of the form p^(3^k), k >= 0, (A186285) and their multiplicative inverses, allows each of those prime powers and their multiplicative inverses to be used at most once, since this corresponds to the balanced ternary representation of the exponents of the prime powers p^a and their multiplicative inverses of the prime factorization of positive rational numbers.

			The balanced ternary digits {-1,0,+1} are represented here as {2,0,1}.
n  num+den          Factors from A186285  Balanced ternary representation
1     2      1 / 1     Empty product                  0
2     3      1 / 2     (1/2)                          2
3     3      2 / 1     2                              1
4     4      1 / 3     (1/3)                          20
5     4      3 / 1     3                              10
6     5      1 / 4     (1/8)*2                        20001
7     5      2 / 3     (1/3)*2                        21
8     5      3 / 2     3*(1/2)                        12
9     5      4 / 1     8*(1/2)                        10002
10    6      1 / 5     (1/5)                          200
11    6      5 / 1     5                              100

OEIS Wiki, Cantor ordering of positive rational numbers

Cf. A020652, A020653, A186285, A186286, A186287, A050376.

A240231 Number of factors needed in the unique factorization of positive integers into terms of A186285 where any term is used at most twice.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 2, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 3, 2, 2, 3, 1, 3, 2, 3, 1, 3, 1, 2, 3, 3, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 2, 3, 2, 2, 2, 4, 1, 3, 3, 4, 1, 3, 1, 2, 3

Offset: 1

Wolfdieter Lang, May 15 2014

The number 1 with factorization defined to be 1 has been included. See a comment on A240230.
This is the row length sequence for the table A240230.
a(n) = 1 if and only if n = 1 or n is a term of A186285.

			a(12) = 3 because the usual prime factorization is 12 = 2^2*3^1 and (2)_3 = [2] and (1)_3 = [1], hence the sum of the base-3 representations of the exponents is 3.
a(24) = 2 as 24 = 3*8, using two factors from A186285. Note also how 3*8 = 3^1 * 2^3, and ternary representations of 1 and 3 are "1" and "10", thus their digit sum is 2. - _Antti Karttunen_, Aug 12 2017
a(36) = 4 from 2^2*3^2, (2)_3 = [2] and 2 + 2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A053735, A077761,A186285, A240230.

Block[{nn = 105, s}, s = Select[Select[Range@ nn, PrimePowerQ], IntegerQ@ Log[3, FactorInteger[#][[1, -1]]] &]; {1}~Join~Table[Length@ Rest@ NestWhileList[Function[{k, m}, {k/#, #} &@ SelectFirst[Reverse@ TakeWhile[s, # <= k &], Divisible[k, #] &]] @@ # &, {n, 1}, First@ # > 1 &][[All, -1]], {n, 2, nn}]] (* Michael De Vlieger, Aug 14 2017 *)
a[n_] := Total[Plus @@ IntegerDigits[#, 3] & /@ (FactorInteger[n][[;; , 2]])]; Array[a, 100] (* Amiram Eldar, May 18 2023 *)

(define (A240231 n) (if (= 1 n) n (A240231with_a1_0 n)))
(definec (A240231with_a1_0 n) (if (= 1 n) 0 (+ (A053735 (A067029 n)) (A240231with_a1_0 (A028234 n)))))
;; Antti Karttunen, Aug 12 2017

a(n) is, for n >= 2, the sum of all entries in the base 3 representation of the exponents of the primes in the usual prime number factorization of n.
From Antti Karttunen, Aug 12 2017: (Start)
That is, apart from the initial term, additive with a(p^e) = A053735(e).
Define b(1) = 0; and for n > 1, b(n) = A053735(A067029(n)) + b(A028234(n)). Then a(n) = b(n) for n > 1, with a(1) = 1 by convention.
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.38090372984518844518..., where f(x) = -x + Sum_{k>=0} (x^(3^k) + 2*x^(2*3^k))/(1 + x^(3^k) + x^(2*3^k)). - Amiram Eldar, Sep 28 2023

Description clarified and more terms added by Antti Karttunen, Aug 12 2017

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 *)

A375270 Numbers of the form p^Fibonacci(2*k), where p is a prime and k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 256

Offset: 1

Amiram Eldar, Aug 09 2024

nonn

Differs from A186285 by having the terms 1, 2^8 = 256, 3^8 = 6561, ..., and not having the terms 2^9 = 512, 3^9 = 19683, ... .

The partial products of this sequence (A375271) are the sequence of numbers with record numbers of Zeckendorf-infinitary divisors (A318465).

			The positive even-indexed Fibonacci numbers are 1, 3, 8, 21, ..., so the sequence includes 2^1 = 2, 2^3 = 8, 2^8 = 256, ..., 3^1 = 3, 3^3 = 27, 3^8 = 6561, ... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A115975.
Subsequences: A000040, A030078, A179645.
Cf. A000045, A001906, A050376, A186285, A318465, A375271 (partial products).

fib[lim_] := Module[{s = {}, f = 1, k = 2}, While[f <= lim, AppendTo[s, f]; k += 2; f = Fibonacci[k]]; s];
seq[max_] := Module[{s = {1}, p = 2, e = 1, f = {}}, While[e > 0, e = Floor[Log[p, max]]; If[f == {}, f = fib[e], f = Select[f, # <= e &]]; s = Join[s, p^f]; p = NextPrime[p]]; Sort[s]]; seq[256]

fib(lim) = {my(s = List(), f = 1, k = 2); while(f <= lim, listput(s, f); k += 2; f = fibonacci(k)); Vec(s);}
lista(pmax) = {my(s = [1], p = 2, e = 1, f = []); while(e > 0, e = logint(pmax, p); if(#f == 0, f = fib(e), f = select(x -> x <= e, f)); s = concat(s, apply(x -> p^x, f)); p = nextprime(p+1)); vecsort(s);}

a(n) = A375271(n)/A375271(n-1) for n >= 2.

cp's OEIS Frontend

A186287 a(n) is the denominator of the rational number whose "factorization" into terms of A186285 has the balanced ternary representation corresponding to n.

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A186286 a(n) is the numerator of the rational number whose "factorization" into terms of A186285 has the balanced ternary representation corresponding to n.

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A185169 Cantor's ordering of positive rational numbers, where a(n) is the balanced ternary representation of the "factorization" of the positive rational number into terms of A186285.

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A240231 Number of factors needed in the unique factorization of positive integers into terms of A186285 where any term is used at most twice.

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

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A375270 Numbers of the form p^Fibonacci(2*k), where p is a prime and k >= 0.

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