A004128 -id:A004128 - cp's OEIS frontend

A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

0, 2, 4, 8, 10, 12, 16, 20, 26, 28, 30, 34, 38, 42, 46, 52, 56, 62, 68, 72, 80, 82, 84, 88, 92, 96, 100, 106, 110, 116, 122, 126, 134, 140, 144, 152, 160, 164, 170, 176, 180, 188, 194, 198, 204, 212, 216, 224, 232, 242, 244, 246, 250, 254, 258, 262, 268, 272, 278, 284, 288, 296, 302, 306, 314, 322, 326, 332, 338, 342, 350, 356, 360

Offset: 0

Antti Karttunen, Sep 11 2016

Antti Karttunen, Table of n, a(n) for n = 0..6250

Cf. A004128, A024023, A053735, A054861, A261231 (left inverse), A261233, A276622, A276624, A276603 (terms divided by 2), A276604 (first differences).
Cf. A179016, A219648, A219666, A255056, A259934, A276573, A276583, A276613 for similar constructions.
Cf. also A263273.

(define (A276623 n) (A276624 (A276622 n)))

a(n) = A276624(A276622(n)).
Other identities. For all n >= 0:
A261231(a(n)) = n.
a(A261233(n)) = A024023(n) = 3^n - 1.

A004117 Numerators of expansion of (1-x)^(-1/3).

Original entry on oeis.org

1, 1, 2, 14, 35, 91, 728, 1976, 5434, 135850, 380380, 1071980, 9111830, 25933670, 74096200, 637227320, 1832028545, 5280552865, 137294374490, 397431084050, 1152550143745, 10043651252635, 29217894553120, 85112997176480

Offset: 0

N. J. A. Sloane

nonn

For n >= 1, a(n) is also the numerator of beta(n+1/3,2/3)*sqrt(27)/(2*Pi). - Groux Roland, May 17 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 93.

Cf. A004128.

Cf. A004987, A007559, A047657, A054861, A034689, A053101, A072888.

Table[Numerator[Binomial[-1/3,n] (-1)^n],{n,0,40}] (* Vincenzo Librandi, Jun 13 2012 *)

a(n)=prod(k=1,n,3*k-2)/n!*3^sum(k=1,n,valuation(k,3))

(1/n!) * 3^A054861(n) * Product_{k=0..n-1} (3k+1). - Ralf Stephan, Mar 13 2004

Numerators in (1-3t)^(-1/3) = 1 + t + 2*t^2 + (14/3)*t^3 + (35/3)*t^4 + (91/3)*t^5 + (728/9)*t^6 + (1976/9)*t^7 + (5434/9)*t^8 + ... = 1 + t + 4*t^2/2! + 28*t^3/3! + 280*t^4/4! + 3640*t^5/5! + 58240*t^6/6! + ... = e.g.f. for triple factorials A007559 (cf. A094638). - Tom Copeland, Dec 04 2013

Typo in formula fixed by Pontus von Brömssen, Nov 25 2008

A268678 Distinct values in A268395; partial sums of A268679.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 8, 11, 15, 16, 18, 19, 20, 22, 23, 26, 27, 31, 32, 34, 36, 37, 40, 41, 42, 47, 48, 50, 52, 53, 56, 57, 59, 60, 64, 65, 66, 69, 70, 72, 74, 75, 81, 82, 83, 86, 87, 89, 90, 92, 93, 98, 101, 102, 104, 105, 106, 108, 109, 113, 116, 117, 119, 120, 121, 123, 124, 127, 131, 132, 134, 135, 136, 138, 139, 142

Offset: 0

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768

Cf. A001969, A268395, A268679.
Cf. A268677 (complement).
Cf. A268680 (least monotonic left inverse).
Cf. A268712.
Cf. also A004128.

f[n_] := Which[n == 1, 0, OddQ@ #, 0, EvenQ@ #, 1 + f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Union@ Accumulate@ Array[f, {150}] (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

a(0) = 0, for n >= 1, a(n) = A268679(n) + a(n-1).
a(n) = A268395(A001969(1+n)).
Other identities. For all n >= 0:
A268680(a(n)) = n.

A061160 Numerators in expansion of Euler transform of b(n) = 1/3.

Original entry on oeis.org

1, 1, 5, 50, 215, 646, 8711, 25475, 105925, 3091270, 11691247, 36809705, 445872155, 1364113925, 5085042010, 50975292560, 183383680088, 588817265695, 19512559194875, 62369303509475, 224877933068647, 2214198452392027, 7686538660149565, 25124342108522750

Offset: 0

Vladeta Jovovic, Apr 17 2001

Denominators of c(n) are 3^d(n), where d(n)=power of 3 in (3*n)!, cf. A004128.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Geoffrey B. Campbell, Some n-space q-binomial theorem extensions and similar identities, arXiv:1906.07526 [math.NT], 2019.
Geoffrey B. Campbell, Continued Fractions for partition generating functions, arXiv:2301.12945 [math.CO], 2023.
Geoffrey B. Campbell, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.
Geoffrey B. Campbell and A. Zujev, Some almost partition theoretic identities, Preprint, 2016.
N. J. A. Sloane, Transforms

Cf. A000712, A061159, A061161.

b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d/3, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 28 2017

c[n_] := c[n] = If[n == 0, 1,
     (1/(3n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
a[n_] := Numerator[c[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 24 2022 *)

Numerators of c(n), where c(n) = (1/(3*n))*Sum_{k=1..n} c(n-k)*sigma(k), n>0, c(0)=1.

A067623 Consider the power series (x+1)^(1/3)=1+x/3-x^2/9+5x^3/81+...; sequence gives denominators of coefficients.

Original entry on oeis.org

1, 3, 9, 81, 243, 729, 6561, 19683, 59049, 1594323, 4782969, 14348907, 129140163, 387420489, 1162261467, 10460353203, 31381059609, 94143178827, 2541865828329, 7625597484987, 22876792454961, 205891132094649, 617673396283947

Offset: 0

Benoit Cloitre, Feb 02 2002

All terms are powers of 3.

Cf. A004128, A046161, A067622 (numerators), A123854.

A067623 := n -> denom(binomial(1/3,n)):
seq(A067623(n), n=0..21); # Peter Luschny, Apr 07 2016

Table[Denominator@ Binomial[1/3, n], {n, 0, 22}] (* Michael De Vlieger, Apr 07 2016 *)

a(n) = 3^A004128(n).
a(n) = 3^n*a(floor(n/3)). - Vladeta Jovovic, Mar 01 2004
a(n) = denominator(binomial(1/3, n)). - Peter Luschny, Apr 07 2016

A344853 a(n) = n minus (sum of digits of n in base 3).

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 4, 4, 4, 8, 8, 8, 10, 10, 10, 12, 12, 12, 16, 16, 16, 18, 18, 18, 20, 20, 20, 26, 26, 26, 28, 28, 28, 30, 30, 30, 34, 34, 34, 36, 36, 36, 38, 38, 38, 42, 42, 42, 44, 44, 44, 46, 46, 46, 52, 52, 52, 54, 54, 54, 56, 56, 56, 60, 60, 60, 62, 62, 62

Offset: 0

Thomas König, May 30 2021

All terms are even.

In all sequences of the form f(n) = n minus (sum of digits of n in base b), every term appears b times consecutively. Here b = 3, hence terms are entries of A346502 repeated 3 times. - Bernard Schott, Jul 21 2021

			a(20) = 20 - (2 + 0 + 2) = 16 because 20 is written as 202 in base 3.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A011371 (in base 2), A066568 (in base 10).

Cf. A004128, A053735, A054861, A346502.

a[n_] := n - Plus @@ IntegerDigits[n, 3]; Array[a, 70, 0] (* Amiram Eldar, May 30 2021 *)

a(n) = n - sumdigits(n, 3); \\ Michel Marcus, Jul 11 2021

a(n) = n - A053735(n).
a(n) = 2 * A054861(n).
a(n) = 2 * A004128(floor(n/3)).
a(3*n) = a(3*n+1) = a(3*n+2).

A051066 Partial sums of A051065.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34

Offset: 0

N. J. A. Sloane and Gary W. Adamson

Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov. 11, 1999.

Cf. A004128, A051065, A051067.

Accumulate[Join[{0}, Mod[Accumulate[Table[IntegerExponent[3*n, 3], {n, 1, 100}]], 2]]] (* Amiram Eldar, Jun 02 2025 *)

a(n) = Sum_{k=0..n} A051065(k). - Wesley Ivan Hurt, May 25 2024

More terms from James Sellers, Dec 11 1999

A051067 A051066 read mod 2.

Original entry on oeis.org

0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1

Offset: 0

N. J. A. Sloane and Gary W. Adamson

Letter from Gary W. Adamson concerning Prouhet-Thue-Morse sequence, Nov. 11, 1999.

Cf. A004128, A051065, A051066.

Mod[Accumulate[Join[{0}, Mod[Accumulate[Table[IntegerExponent[3*n, 3], {n, 1, 100}]], 2]]], 2] (* Amiram Eldar, Jun 02 2025 *)

More terms from James Sellers, Dec 11 1999

A127427 a(n) = v_3( (6n)! ) - v_3( (3n)! ), where v_3(N) is the 3-adic valuation A007949(N).

Original entry on oeis.org

0, 1, 3, 4, 5, 8, 9, 10, 12, 13, 14, 16, 17, 18, 22, 23, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 39, 40, 41, 43, 44, 45, 48, 49, 50, 52, 53, 54, 56, 57, 58, 63, 64, 65, 67, 68, 69, 71, 72, 73, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 90, 91, 93, 94, 95, 97, 98, 99, 103, 104, 105, 107

Offset: 0

N. J. A. Sloane, Apr 02 2007

nonn

Sung-Hyuk Cha, On Integer Sequences Derived from Balanced k-ary Trees, Applied Mathematics in Electrical and Computer Engineering, 2012. - From _N. J. A. Sloane_, Jun 12 2012
Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From _N. J. A. Sloane_, Dec 24 2012

Cf. A007949, A053735, A054861, A004128.

Essentially partial sums of A127427.

s[n_] := Plus @@ IntegerDigits[n, 3]; a[n_] := (3*n + s[3*n] - s[6*n])/2; Array[a, 100, 0] (* Amiram Eldar, Feb 21 2021 *)

a(n) = valuation((6*n)!, 3) - valuation((3*n)!, 3); \\ Michel Marcus, Jul 29 2017

a(n) - n = a( [(n+1)/3] ).

a(n) = (3*n + A053735(n) - A053735(6*n))/2. - Amiram Eldar, Feb 21 2021

A381973 Numbers m such that Sum_{k >= 0} floor(m/3^k) is prime.

Original entry on oeis.org

2, 4, 9, 12, 14, 17, 22, 28, 36, 41, 42, 46, 49, 61, 66, 69, 71, 73, 86, 89, 94, 101, 102, 107, 110, 113, 121, 129, 131, 134, 143, 151, 153, 155, 158, 169, 173, 177, 181, 187, 190, 211, 214, 223, 227, 235, 238, 250, 254, 257, 274, 281, 282, 289, 295, 301

Offset: 1

Clark Kimberling, Apr 01 2025

nonn

Includes 3^(k-1) for k in A028491. - Robert Israel, Apr 21 2025

			[9/1] + [9/3] + [9/9] = 13, where [ ] = floor, so 9 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A028491, A381974.

f:= proc(n) add(floor(n/3^k),k=0..ilog[3](n)) end proc:
select(m -> isprime(f(n)), [$2..1000]); # Robert Israel, Apr 21 2025

f[n_] := Sum[Floor[n/3^k], {k, 0, Floor[Log[3, n]]}]  (* A004128 *)
u = Select[Range[400], PrimeQ[f[#]] &]  (* A381973 *)
Map[f, u]   (* A381974 *)

cp's OEIS Frontend

A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

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A004117 Numerators of expansion of (1-x)^(-1/3).

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A268678 Distinct values in A268395; partial sums of A268679.

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A061160 Numerators in expansion of Euler transform of b(n) = 1/3.

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A067623 Consider the power series (x+1)^(1/3)=1+x/3-x^2/9+5x^3/81+...; sequence gives denominators of coefficients.

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A344853 a(n) = n minus (sum of digits of n in base 3).

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A051066 Partial sums of A051065.

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A051067 A051066 read mod 2.

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