A373216 -id:A373216 - cp's OEIS frontend

A373297 Euler transform of A373216.

1, 1, 2, 3, 5, 7, 12, 16, 24, 33, 47, 63, 90, 118, 161, 212, 283, 367, 487, 624, 812, 1037, 1332, 1685, 2152, 2700, 3409, 4259, 5333, 6617, 8242, 10165, 12568, 15436, 18970, 23178, 28360, 34487, 41970, 50850, 61599, 74322, 89696, 107809, 129572, 155235, 185881, 221936

Offset: 0

Seiichi Manyama, May 31 2024

nonn

Cf. A092119, A173241, A373295, A373296, A373298.

Cf. A000041, A373216, A373220.

my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(valuation(k, 6)+1)))

G.f.: A(x) = 1/Product_{i>=1, j>=0} (1 - x^(i * 6^j)).

Let A(x) be the g.f. of this sequence, and P(x) be the g.f. of A000041, then P(x) = A(x)/A(x^6).

A051064 3^a(n) exactly divides 3n. Or, 3-adic valuation of 3n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2

Offset: 1

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

a(n) is the Hamming distance between n and n-1 in ternary representation. - Philippe Deléham, Mar 29 2004
3^a(n) divides 4^n-1. - Benoit Cloitre, Oct 25 2004
Generalized Ruler Function for k=3. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)
a(A007417(n)) is odd and a(A145204(n)) is even. - Reinhard Zumkeller, May 23 2013
First n terms comprise least cubefree word of length n using positive integers, where "cubefree" means that the word contains no three consecutive identical subwords; e.g., 1 contains no cube; 11 contains no cube; 111 does but 112 does not; ... 1,1,2,1,1,2,1,1,1 does, and 1,1,2,1,1,2,1,1,2 does, but 1,1,2,1,1,2,1,1,3 does not, etc. - Clark Kimberling, Sep 10 2013
The sequence is invariant under the "lower trim" operator: remove all ones, and subtract one from each remaining term. - Franklin T. Adams-Watters, May 25 2017
a(n) is the dimension in which the coordinates of the vertices n-1 and n differ in the ternary reflected Gray code. - Arie Bos, Jul 12 2023
The number of powers of 3 that divide n. - Amiram Eldar, Mar 29 2025

			3^2 | 3*6 = 18, so a(6) = 2.

Letter from Gary W. Adamson to N. J. A. Sloane concerning Prouhet-Thue-Morse sequence, Nov. 11, 1999.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 243. Book's website
Tamas Lengyel, Divisiblity Properties by Multisection, Fib. Quart. 41 (1) (2003) 72.
Simon Plouffe, On the values of the functions zeta and gamma, arXiv preprint arXiv:1310.7195 [math.NT], 2013.
Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (The A indices in P at equations 1' and 2' for p=3.)
Index entries for sequences that are fixed points of mappings.

Cf. A007949.
Partial sums give A004128.
Cf. A000005, A027746.
Cf. A254046.
Cf. A001511, A055457, A115362, A373216, A373217.

a051064 = (+ 1) . length .
                  takeWhile (== 3) . dropWhile (== 2) . a027746_row
-- Reinhard Zumkeller, May 23 2013

seq(1+padic:-ordp(n,3), n=1..100); # Robert Israel, Aug 07 2014

Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 1, 2}, 2 -> {1, 1, 3}, 3 -> {1, 1, 4}, 4 -> {1, 1, 5}})]}], {1}, 5] (* Robert G. Wilson v, Mar 03 2005 *)
Table[ IntegerExponent[3n, 3], {n, 1, 105}] (* Jean-François Alcover, Oct 10 2011 *)

PARI
```
a(n)=if(n<1,0,1+valuation(n,3))
```

def A051064(n):
    c = 1
    a, b = divmod(n,3)
    while b == 0:
        a, b = divmod(a,3)
        c += 1
    return c # Chai Wah Wu, Apr 18 2022

a(n) = A007949(n) + 1 = A004128(n) - A004128(n-1).
Multiplicative with a(p^e) = e+1 if p = 3; 1 if p <> 3. - Vladeta Jovovic, Aug 24 2002
G.f.: Sum_{k>=0} x^3^k/(1-x^3^k). - Ralf Stephan, Apr 12 2002
Fixed point of the morphism: 1 -> 112; 2 -> 113; 3 -> 114; 4 -> 115; ...; starting from a(1) = 1. a(3n+1) = a(3n+2) = 1; a(3n) = 1 + a(n). - Philippe Deléham, Mar 29 2004
a(n) = (-1)*Sum_{d divides n} mu(3d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
Dirichlet g.f.: zeta(s)/(1-1/3^s). - R. J. Mathar, Jun 13 2011
a(n) = (1/2)*(3 - A053735(n) + A053735(n-1)) for n >= 1. - Tom Edgar, Aug 06 2014
a(n) = A007949(3n). - Cyril Damamme, Aug 04 2015
a(2n) = a(n), a(2n-1) = A254046(n). - Cyril Damamme, Aug 04 2015
G.f. A(x) satisfies: A(x) = A(x^3) + x/(1 - x). - Ilya Gutkovskiy, May 03 2019
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Sep 11 2020 [corrected by Vaclav Kotesovec, Jun 25 2024, see also A004128]
a(n) = tau(n)/(tau(3*n) - tau(n)), where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
G.f.: Sum_{i>=1, j>=0} x^(i*3^j). - Seiichi Manyama, Mar 23 2025
Conjecture: a(n) = A007949(A000045(4*n)), all other 3-adic quadrisections A007949(A000045(.))=0. [Lengyel?]. - R. J. Mathar, Jun 28 2025

More terms from James Sellers, Dec 11 1999

More terms from Vladeta Jovovic, Aug 24 2002

A055457 5^a(n) exactly divides 5n. Or, 5-adic valuation of 5n.

Original entry on oeis.org

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

Offset: 1

Alford Arnold, Jun 25 2000

More generally, consider the sequence defined by p^a(n) exactly divides p*n. For p = 3 we have A051064 and for p = 2 we have A001511.

The number of powers of 5 that divide n. - Amiram Eldar, Mar 29 2025

			a(5) = 2 since 5^2 exactly divides 5 times 5;
a(25) = 3 since 5^3 exactly divides 5 times 25;
a(125) = 4 since 5^4 exactly divides 5 times 125.

T. D. Noe, Table of n, a(n) for n=1..1000
Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (The A indices in P at equations 1' and 2' for p=5.)

Cf. A007949, A112765, A191610 (partial sums).

Cf. A001511, A051064, A115362, A373216, A373217.

seq(padic:-ordp(5*n,5), n=1..1000); # Robert Israel, Dec 07 2015

max = 1000; s = (1/x)*Sum[x^(5^k)/(1-x^5^k), {k, 0, Log[5, max] // Ceiling }] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 04 2015 *)
Table[IntegerExponent[n, 5] + 1, {n, 1, 100}] (* Amiram Eldar, Sep 21 2020 *)

a(n)=-sumdiv(n,d,moebius(5*d)*numdiv(n/d)) \\ Benoit Cloitre, Jun 21 2007

a(n)=valuation(5*n,5) \\ Anders Hellström, Dec 04 2015

def A055457(n):
    c = 1
    while not (a:=divmod(n,5))[1]:
        c += 1
        n = a[0]
    return c # Chai Wah Wu, Feb 28 2025

G.f.: Sum_{k>=0} x^(5^k)/(1-x^5^k). - Ralf Stephan, Apr 12 2002
Multiplicative with a(p^e) = e+1 if p = 5, 1 otherwise.
a(n) = -Sum_{d|n} mu(5d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
Dirichlet g.f.: zeta(s)/(1-1/5^s). - R. J. Mathar, Feb 09 2011
a(n) = A112765(5n). - R. J. Mathar, Jul 17 2012
a(5n) = 1 + a(n).  a(5n+k) = 1 for k = 1..4. - Robert Israel, Dec 07 2015
G.f. satisfies A(x^5) = A(x) - x/(1-x). - Robert Israel, Dec 08 2015
a(n) = A112765(n) + 1. - Amiram Eldar, Sep 21 2020
Sum_{k=1..n} a(k) ~ 5*n/4. - Vaclav Kotesovec, Sep 21 2020
G.f.: Sum_{i>=1, j>=0} x^(i*5^j). - Seiichi Manyama, Mar 23 2025

A373217 Expansion of Sum_{k>=0} x^(7^k) / (1 - x^(7^k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, May 28 2024

The number of powers of 7 that divide n. - Amiram Eldar, Mar 29 2025

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001511, A051064, A055457, A115362, A373216.

Cf. A000005, A008683, A214411, A373221.

a[n_] := 1 + IntegerExponent[n, 7]; Array[a, 100] (* Amiram Eldar, May 29 2024 *)

PARI
```
a(n) = valuation(n, 7)+1;
```

G.f. A(x) satisfies A(x) = x/(1 - x) + A(x^7).
a(7*n+1) = a(7*n+2) = ... = (7*n+6) = 1 and a(7*n+7) = 1 + a(n+1) for n >= 0.
Multiplicative with a(p^e) = e+1 if p = 7, 1 otherwise.
a(n) = -Sum_{d|n} mu(7*d) * tau(n/d).
a(n) = A214411(n) + 1.
From Amiram Eldar, May 29 2024: (Start)
Dirichlet g.f.: (7^s/(7^s-1)) * zeta(s).
Sum_{k=1..n} a(k) ~ (7/6) * n. (End)
G.f.: Sum_{i>=1, j>=0} x^(i*7^j). - Seiichi Manyama, Mar 23 2025
a(n) = A214411(7*n). - R. J. Mathar, Jun 28 2025

A373220 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 6^j)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 7, 10, 12, 15, 20, 24, 29, 37, 44, 53, 67, 79, 94, 115, 135, 160, 193, 226, 265, 315, 367, 428, 505, 585, 678, 792, 913, 1054, 1225, 1406, 1614, 1862, 2129, 2436, 2797, 3187, 3630, 4147, 4709, 5347, 6084, 6887, 7793, 8832, 9968, 11247, 12706, 14301, 16089, 18116, 20337

Offset: 0

Seiichi Manyama, May 28 2024

nonn

Cf. A000041, A174065, A327726, A373219, A373221.

Cf. A000009, A373216.

my(N=60, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(valuation(k, 6)+1)))

G.f.: Product_{k>=1} (1 + x^k)^A373216(k).

Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^6).

A373282 Expansion of Sum_{k>=0} x^(6^k) / (1 - 6*x^(6^k)).

Original entry on oeis.org

1, 6, 36, 216, 1296, 7777, 46656, 279936, 1679616, 10077696, 60466176, 362797062, 2176782336, 13060694016, 78364164096, 470184984576, 2821109907456, 16926659444772, 101559956668416, 609359740010496, 3656158440062976, 21936950640377856

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373279, A373280, A373281, A373283.

Cf. A000400, A373216.

G.f. A(x) satisfies A(x) = x/(1 - 6*x) + A(x^6).

If n == 0 (mod 6), a(n) = 6^n + a(n/6) otherwise a(n) = 6^n.

A382378 Expansion of 1/( 1 - Sum_{k>=0} x^(6^k) / (1 - x^(6^k)) ).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 33, 66, 133, 268, 540, 1088, 2194, 4421, 8910, 17957, 36190, 72936, 146996, 296252, 597061, 1203306, 2425121, 4887544, 9850272, 19852060, 40009486, 80634401, 162509126, 327517977, 660073866, 1330301036, 2681064864, 5403370072, 10889855193, 21947218962

Offset: 0

Seiichi Manyama, Mar 23 2025

nonn

Cf. A327736, A382367, A382372, A382373.

Cf. A122841, A373216.

a(0) = 1; a(n) = Sum_{k=1..n} A373216(k) * a(n-k).
G.f.: 1/(1 - Sum_{i>=1, j>=0} x^(i*6^j)).
G.f. A(x) satisfies A(x) = 1/( 1/A(x^6) - x/(1-x) ).

A373397 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 10, 11, 14, 13, 14, 15, 16, 17, 21, 19, 20, 21, 22, 23, 28, 25, 26, 27, 28, 29, 35, 31, 32, 33, 34, 35, 43, 37, 38, 39, 40, 41, 49, 43, 44, 45, 46, 47, 56, 49, 50, 51, 52, 53, 63, 55, 56, 57, 58, 59, 70, 61, 62, 63, 64, 65, 77, 67, 68, 69, 70, 71, 86, 73, 74, 75, 76, 77, 91

Offset: 1

Seiichi Manyama, Jun 04 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A129527, A327625, A359099, A359100, A373188.

Cf. A373216.

G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^6).

If n == 0 (mod 6), a(n) = n + a(n/6) otherwise a(n) = n.

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A373297 Euler transform of A373216.

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A051064 3^a(n) exactly divides 3n. Or, 3-adic valuation of 3n.

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A055457 5^a(n) exactly divides 5n. Or, 5-adic valuation of 5n.

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A373217 Expansion of Sum_{k>=0} x^(7^k) / (1 - x^(7^k)).

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A373220 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 6^j)).

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A373282 Expansion of Sum_{k>=0} x^(6^k) / (1 - 6*x^(6^k)).

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A382378 Expansion of 1/( 1 - Sum_{k>=0} x^(6^k) / (1 - x^(6^k)) ).

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A373397 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k))^2.

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