A290568 -id:A290568 - cp's OEIS frontend

A290563 Coefficients in 5-adic expansion of 3^(1/3).

2, 2, 3, 1, 0, 3, 3, 2, 1, 2, 0, 0, 0, 2, 3, 4, 3, 3, 4, 4, 0, 3, 2, 0, 2, 2, 2, 2, 0, 1, 0, 3, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 2, 3, 3, 1, 4, 0, 3, 4, 4, 3, 1, 3, 2, 4, 4, 3, 0, 3, 3, 1, 1, 2, 0, 3, 0, 2, 3, 4, 3, 0, 2, 2, 2, 3, 1, 4, 3, 2, 0, 1, 0, 3, 3, 4

Offset: 0

Seiichi Manyama, Aug 06 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A290566, A290568.

Vecrev( digits( truncate( (3+O(5^100))^(1/3) ), 5) ) \\ Joerg Arndt, Aug 06 2017

require 'OpenSSL'
def f_a(ary, a)
  (0..ary.size - 1).inject(0){|s, i| s + ary[i] * a ** i}
end
def df(ary)
  (1..ary.size - 1).map{|i| i * ary[i]}
end
def A(c_ary, k, m, n)
  x = OpenSSL::BN.new((-f_a(df(c_ary), k)).to_s).mod_inverse(m).to_i % m
  f_ary = c_ary.map{|i| x * i}
  f_ary[1] += 1
  d_ary = []
  ary = [0]
  a, mod = k, m
  (n + 1).times{|i|
    b = a % mod
    d_ary << (b - ary[-1]) / m ** i
    ary << b
    a = f_a(f_ary, b)
    mod *= m
  }
  d_ary
end
def A290563(n)
  A([-3, 0, 0, 1], 2, 5, n)
end
p A290563(100)

A290567 The successive approximations up to 5^n for 5-adic integer 2^(1/3).

Original entry on oeis.org

0, 3, 3, 53, 303, 2178, 5303, 67803, 67803, 849053, 6708428, 6708428, 6708428, 983270928, 983270928, 25397333428, 147467645928, 605231317803, 1368170770928, 5182868036553, 43329840692803, 43329840692803, 43329840692803, 4811701422724053, 52495417243036553

Offset: 0

Seiichi Manyama, Aug 06 2017

nonn

x   = ...132203,
x^2 = ...344214,
x^3 = ...000002 = 2.

			a(1) = (   3)_5 = 3,
a(2) = (   3)_5 = 3,
a(3) = ( 203)_5 = 53,
a(4) = (2203)_5 = 303.

Seiichi Manyama, Table of n, a(n) for n = 0..1431
Wikipedia, Hensel's Lemma.

Cf. A290566, A290568.

a(n)=truncate((2+O(5^n))^(1/3)); \\ Joerg Arndt, Aug 06 2017

a(0) = 0 and a(1) = 3, a(n) = a(n-1) + 2 * (a(n-1)^3 - 2) mod 5^n for n > 1.

A320914 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 7 (mod 13) case (except for n = 0).

Original entry on oeis.org

0, 7, 7, 1021, 20794, 77916, 4533432, 57628331, 810610535, 8967917745, 40781415864, 592215383260, 22098140111704, 208482821091552, 3842984100198588, 23529866028695033, 586574689183693360, 5244490953465952247, 74447818308516655711, 524269446116346228227, 9295791188369022892289

Offset: 0

Jianing Song, Aug 27 2019

nonn

For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 7 mod 13 such that k^3 - 5 is divisible by 13^n.

For k not divisible by 13, k is a cube in 13-adic field if and only if k == 1, 5, 8, 12 (mod 13). If k is a cube in 13-adic field, then k has exactly three cubic roots.

			The unique number k in [1, 13^2] and congruent to 7 modulo 13 such that k^3 - 5 is divisible by 13^2 is k = 7, so a(2) = 7.
The unique number k in [1, 13^3] and congruent to 7 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 1021, so a(3) = 1021.

Wikipedia, p-adic number

Cf. A320915, A321105, A321106, A321107, A321108.

For 5-adic cubic roots, see A290567, A290568, A309444.

a(n) = lift(sqrtn(5+O(13^n), 3) * (-1+sqrt(-3+O(13^n)))/2)

A320915 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 8 (mod 13) case (except for n = 0).

Original entry on oeis.org

0, 8, 8, 177, 11162, 211089, 211089, 24345134, 777327338, 7303173106, 113348166836, 1629791577175, 12382753941397, 222065520043726, 1130690839820485, 16880196382617641, 272809661453071426, 5596142534918510154, 14246558454299848087, 576523593214086813732, 4962284464340425145763

Offset: 0

Jianing Song, Aug 27 2019

nonn

For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 8 mod 13 such that k^3 - 5 is divisible by 13^n.

For k not divisible by 13, k is a cube in 13-adic field if and only if k == 1, 5, 8, 12 (mod 13). If k is a cube in 13-adic field, then k has exactly three cubic roots.

			The unique number k in [1, 13^2] and congruent to 8 modulo 13 such that k^3 - 5 is divisible by 13^2 is k = 8, so a(2) = 8.
The unique number k in [1, 13^3] and congruent to 8 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 177, so a(3) = 177.

Wikipedia, p-adic number

Cf. A320914, A321105, A321106, A321107, A321108.

For 5-adic cubic roots, see A290567, A290568, A309444.

PARI
```
a(n) = lift(sqrtn(5+O(13^n), 3))
```

A321105 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 11 (mod 13) case (except for n = 0).

Original entry on oeis.org

0, 11, 154, 999, 25166, 82288, 82288, 43523569, 43523569, 4937907895, 121587400998, 1362313827639, 12115276191861, 175201872049228, 2901077831379505, 10775830602778083, 471448867729594896, 6460198350378213465, 23761030189140889331, 361127251045013068718, 4746888122171351400749

Offset: 0

Jianing Song, Aug 27 2019

nonn

For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 11 mod 13 such that k^3 - 5 is divisible by 13^n.

For k not divisible by 13, k is a cube in 13-adic field if and only if k == 1, 5, 8, 12 (mod 13). If k is a cube in 13-adic field, then k has exactly three cubic roots.

			The unique number k in [1, 13^2] and congruent to 11 modulo 13 such that k^3 - 5 is divisible by 13^2 is k = 154, so a(2) = 154.
The unique number k in [1, 13^3] and congruent to 11 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 999, so a(3) = 999.

Wikipedia, p-adic number

Cf. A320914, A320915, A321106, A321107, A321108.

For 5-adic cubic roots, see A290567, A290568, A309444.

a(n) = lift(sqrtn(5+O(13^n), 3) * (-1-sqrt(-3+O(13^n)))/2)

A309444 The successive approximations up to 5^n for 5-adic integer 4^(1/3).

Original entry on oeis.org

0, 4, 9, 59, 559, 3059, 12434, 59309, 371809, 371809, 8184309, 27715559, 76543684, 320684309, 1541387434, 25955449934, 86990606184, 392166387434, 2680984746809, 14125076543684, 52272049199934, 338374344121809, 2245722976934309, 7014094558965559, 42776881424199934

Offset: 0

Seiichi Manyama, Aug 03 2019

nonn

			a(1) = (   4)_5 = 4,
a(2) = (  14)_5 = 9,
a(3) = ( 214)_5 = 59,
a(4) = (4214)_5 = 559.

Cf. A309443.
Expansions of p-adic integers:
A268922, A269590 (5-adic, sqrt(-4));
A048898, A048899 (5-adic, sqrt(-1));
A290567 (5-adic, 2^(1/3));
A290568 (5-adic, 3^(1/3)).

PARI
```
{a(n) = truncate((4+O(5^n))^(1/3))}
```

a(0) = 0 and a(1) = 4, a(n) = a(n-1) + 3 * (a(n-1)^3 - 4) mod 5^n for n > 1.

A319097 One of the three successive approximations up to 7^n for 7-adic integer 6^(1/3). This is the 3 (mod 7) case (except for n = 0).

Original entry on oeis.org

0, 3, 24, 122, 808, 10412, 111254, 817148, 1640691, 24699895, 186114323, 186114323, 6118094552, 6118094552, 490563146587, 2525232365134, 26263039914849, 59495970484450, 1222648540420485, 6107889334151832, 74501260446390690, 234085793041614692, 1351177521208182706, 24810103812706111000, 134285093173029776372

Offset: 0

Jianing Song, Aug 27 2019

nonn

For n > 0, a(n) is the unique number k in [1, 7^n] and congruent to 3 mod 7 such that k^3 - 6 is divisible by 7^n.

For k not divisible by 7, k is a cube in 7-adic field if and only if k == 1, 6 (mod 7). If k is a cube in 7-adic field, then k has exactly three cubic roots.

			The unique number k in [1, 7^2] and congruent to 3 modulo 7 such that k^3 - 6 is divisible by 7^2 is k = 24, so a(2) = 24.
The unique number k in [1, 7^3] and congruent to 3 modulo 7 such that k^3 - 6 is divisible by 7^3 is k = 122, so a(3) = 122.

Wikipedia, p-adic number

Cf. A319297, A319305, A319555.
Approximations of p-adic cubic roots:
A290567 (5-adic, 2^(1/3));
A290568 (5-adic, 3^(1/3));
A309444 (5-adic, 4^(1/3));
this sequence, A319098, A319199 (7-adic, 6^(1/3));
A320914, A320915, A321105 (13-adic, 5^(1/3)).

a(n) = lift(sqrtn(6+O(7^n), 3) * (-1+sqrt(-3+O(7^n)))/2)

a(n) = A319098(n)*(A210852(n)-1) mod 7^n = A319098(n)*A210852(n)^2 mod 7^n.

a(n) = A319199(n)*(A212153(n)-1) mod 7^n = A319199(n)*A212153(n)^2 mod 7^n.

A319098 One of the three successive approximations up to 7^n for 7-adic integer 6^(1/3). This is the 5 (mod 7) case (except for n = 0).

Original entry on oeis.org

0, 5, 40, 138, 824, 3225, 87260, 793154, 793154, 29617159, 191031587, 1320932583, 7252912812, 7252912812, 7252912812, 2041922131359, 16284606661188, 82750467800390, 1013272523749218, 9155340513301463, 31953130884047749, 111745397181659750, 670291261264943757

Offset: 0

Jianing Song, Aug 27 2019

nonn

For n > 0, a(n) is the unique number k in [1, 7^n] and congruent to 5 mod 7 such that k^3 - 6 is divisible by 7^n.

For k not divisible by 7, k is a cube in 7-adic field if and only if k == 1, 6 (mod 7). If k is a cube in 7-adic field, then k has exactly three cubic roots.

			The unique number k in [1, 7^2] and congruent to 5 modulo 7 such that k^3 - 6 is divisible by 7^2 is k = 40, so a(2) = 40.
The unique number k in [1, 7^3] and congruent to 5 modulo 7 such that k^3 - 6 is divisible by 7^3 is k = 138, so a(3) = 138.

Wikipedia, p-adic number

Cf. A319297, A319305, A319555.
Approximations of p-adic cubic roots:
A290567 (5-adic, 2^(1/3));
A290568 (5-adic, 3^(1/3));
A309444 (5-adic, 4^(1/3));
A319097, this sequence, A319199 (7-adic, 6^(1/3));
A320914, A320915, A321105 (13-adic, 5^(1/3)).

a(n) = lift(sqrtn(6+O(7^n), 3) * (-1-sqrt(-3+O(7^n)))/2)

a(n) = A319097(n)*(A212153(n)-1) mod 7^n = A319097(n)*A212153(n)^2 mod 7^n.

a(n) = A319199(n)*(A210852(n)-1) mod 7^n = A319199(n)*A210852(n)^2 mod 7^n.

A319199 One of the three successive approximations up to 7^n for 7-adic integer 6^(1/3). This is the 6 (mod 7) case (except for n = 0).

Original entry on oeis.org

0, 6, 34, 83, 769, 3170, 36784, 36784, 3330956, 26390160, 187804588, 470279837, 470279837, 83518003043, 180407013450, 180407013450, 23918214563165, 90384075702367, 1020906131651195, 7534560523292991, 53130141264785563, 212714673860009565, 1888352266109861586

Offset: 0

Jianing Song, Aug 27 2019

nonn

For n > 0, a(n) is the unique number k in [1, 7^n] and congruent to 6 mod 7 such that k^3 - 6 is divisible by 7^n.

For k not divisible by 7, k is a cube in 7-adic field if and only if k == 1, 6 (mod 7). If k is a cube in 7-adic field, then k has exactly three cubic roots.

			The unique number k in [1, 7^2] and congruent to 6 modulo 7 such that k^3 - 6 is divisible by 7^2 is k = 34, so a(2) = 24.
The unique number k in [1, 7^3] and congruent to 6 modulo 7 such that k^3 - 6 is divisible by 7^3 is k = 83, so a(3) = 122.

Wikipedia, p-adic number

Cf. A319297, A319305, A319555.
Approximations of p-adic cubic roots:
A290567 (5-adic, 2^(1/3));
A290568 (5-adic, 3^(1/3));
A309444 (5-adic, 4^(1/3));
A319097, A319098, this sequence (7-adic, 6^(1/3));
A320914, A320915, A321105 (13-adic, 5^(1/3)).

PARI
```
a(n) = lift(sqrtn(6+O(7^n), 3))
```

a(n) = A319097(n)*(A210852(n)-1) mod 7^n = A319097(n)*A210852(n)^2 mod 7^n.

a(n) = A319098(n)*(A212153(n)-1) mod 7^n = A319098(n)*A212153(n)^2 mod 7^n.

A309451 The successive approximations up to 7^n for 7-adic integer 3^(1/5).

Original entry on oeis.org

0, 5, 26, 75, 1104, 3505, 20312, 20312, 4961570, 28020774, 229788809, 512264058, 2489590801, 71696026806, 71696026806, 71696026806, 19061942066578, 218459525484184, 451090039471391

Offset: 0

Seiichi Manyama, Aug 03 2019

nonn

			a(1) = (   5)_7 = 5,
a(2) = (  35)_7 = 26,
a(3) = ( 135)_7 = 75,
a(4) = (3135)_7 = 1104.

Cf. A309446.
Expansions of p-adic integers:
A290568 (5-adic, 3^(1/3));
A290800, A290802 (7-adic, sqrt(-6));
A290806, A290809 (7-adic, sqrt(-5));
A290803, A290804 (7-adic, sqrt(-3));
A210852, A212153 (7-adic, (1+sqrt(-3))/2);
A290557, A290559 (7-adic, sqrt(2));
A309450 (7-adic, 2^(1/5));
A309452 (7-adic, 4^(1/5));
A309453 (7-adic, 5^(1/5));
A309454 (7-adic, 6^(1/5)).

PARI
```
{a(n) = truncate((3+O(7^n))^(1/5))}
```

a(0) = 0 and a(1) = 5, a(n) = a(n-1) + 2 * (a(n-1)^5 - 3) mod 7^n for n > 1.

cp's OEIS Frontend

A290563 Coefficients in 5-adic expansion of 3^(1/3).

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A290567 The successive approximations up to 5^n for 5-adic integer 2^(1/3).

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A320914 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 7 (mod 13) case (except for n = 0).

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A320915 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 8 (mod 13) case (except for n = 0).

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A321105 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 11 (mod 13) case (except for n = 0).

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A309444 The successive approximations up to 5^n for 5-adic integer 4^(1/3).

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A319097 One of the three successive approximations up to 7^n for 7-adic integer 6^(1/3). This is the 3 (mod 7) case (except for n = 0).

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A319098 One of the three successive approximations up to 7^n for 7-adic integer 6^(1/3). This is the 5 (mod 7) case (except for n = 0).

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