A290803 -id:A290803 - cp's OEIS frontend

A309450 The successive approximations up to 7^n for 7-adic integer 2^(1/5).

0, 4, 46, 95, 1124, 15530, 82758, 435705, 4553420, 27612624, 269734266, 1682110511, 9591417483, 9591417483, 9591417483, 4078929854577, 23069175894349, 122767967603152, 1053290023551980, 9195358013104225, 77588729125343083, 237173261720567085, 1354264989887135099

Offset: 0

Seiichi Manyama, Aug 03 2019

nonn

			a(1) = (   4)_7 = 4,
a(2) = (  64)_7 = 46,
a(3) = ( 164)_7 = 95,
a(4) = (3164)_7 = 1124.

Robert Israel, Table of n, a(n) for n = 0..1182

Cf. A309445.
Expansions of p-adic integers:
A290567 (5-adic, 2^(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));
A309451 (7-adic, 3^(1/5));
A309452 (7-adic, 4^(1/5));
A309453 (7-adic, 5^(1/5));
A309454 (7-adic, 6^(1/5)).

A:= op([1,3],padic:-rootp(x^5 -2,  7, 25)):
seq(add(A[i]*10^(i-1),i=1..n),n=0..25); # Robert Israel, Aug 04 2019

PARI
```
{a(n) = truncate((2+O(7^n))^(1/5))}
```

a(0) = 0 and a(1) = 4, a(n) = a(n-1) + (a(n-1)^5 - 2) mod 7^n for n > 1.

A290804 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 5 mod 7 (except for the initial 0).

Original entry on oeis.org

0, 5, 12, 306, 306, 2707, 69935, 658180, 4775895, 10540696, 10540696, 575491194, 4530144680, 59895293484, 544340345519, 1900786491217, 20891032530989, 87356893670191, 319987407657398, 10090468995120092, 44287154551239521, 203871687146463523

Offset: 0

Seiichi Manyama, Aug 11 2017

x = ...410615,

x^2 = ...666664 = -3.

			a(1) =     5_7 = 5,
a(2) =    15_7 = 12,
a(3) =   615_7 = 306,
a(4) =   615_7 = 306,
a(5) = 10615_7 = 2707.

Seiichi Manyama, Table of n, a(n) for n = 0..1183
Peter Bala, Using Lucas polynomials to find the p -adic square roots of -1, -2 and -3, Dec 2022.
Wikipedia, Hensel's Lemma.

Cf. A114525, A290797, A290803.

a(n) = if (n, 7^n - truncate(sqrt(-3+O(7^(n)))), 0)

a(0) = 0 and a(1) = 5, a(n) = a(n-1) + 2 * (a(n-1)^2 + 3) mod 7^n for n > 1.
If n > 0, a(n) = 7^n - A290803(n).
a(n) = L(7^n,5) (mod 7^n) = ( ((5 + sqrt(29))/2)^(7^n) + ((5 - sqrt(29))/2)^(7^n) ) (mod 7^n), where L(n,x) denotes the n-th Lucas polynomial of A114525. - Peter Bala, Nov 28 2022

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.

A309452 The successive approximations up to 7^n for 7-adic integer 4^(1/5).

Original entry on oeis.org

0, 2, 9, 107, 450, 450, 67678, 655923, 2303009, 13832611, 54186218, 1749037712, 13612998170, 27454285371, 124343295778, 4193681732872, 18436366262701, 217833949680307, 1380986519616342, 3009400117526791, 3009400117526791, 162593932712750793, 3513869117212454835

Offset: 0

Seiichi Manyama, Aug 03 2019

nonn

			a(1) = (   2)_7 = 2,
a(2) = (  12)_7 = 9,
a(3) = ( 212)_7 = 107,
a(4) = (1212)_7 = 450.

Cf. A309447.
Expansions of p-adic integers:
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));
A309451 (7-adic, 3^(1/5));
A309453 (7-adic, 5^(1/5));
A309454 (7-adic, 6^(1/5)).

PARI
```
{a(n) = truncate((4+O(7^n))^(1/5))}
```

a(0) = 0 and a(1) = 2, a(n) = a(n-1) + 2 * (a(n-1)^5 - 4) mod 7^n for n > 1.

A309453 The successive approximations up to 7^n for 7-adic integer 5^(1/5).

Original entry on oeis.org

0, 3, 45, 339, 1368, 8571, 42185, 630430, 4748145, 27807349, 27807349, 1722658843, 13586619301, 41269193703, 235047214517, 2269716433064, 30755085492722, 230152668910328, 928044210871949, 2556457808782398, 36753143364901827, 196337675960125829, 2430521132293261857

Offset: 0

Seiichi Manyama, Aug 03 2019

nonn

			a(1) = (   3)_7 = 3,
a(2) = (  63)_7 = 45,
a(3) = ( 663)_7 = 339,
a(4) = (3663)_7 = 1368.

Cf. A309448.
Expansions of p-adic integers:
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));
A309451 (7-adic, 3^(1/5));
A309452 (7-adic, 4^(1/5));
A309454 (7-adic, 6^(1/5)).

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

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

A309454 The successive approximations up to 7^n for 7-adic integer 6^(1/5).

Original entry on oeis.org

0, 6, 20, 265, 1980, 11584, 11584, 246882, 1070425, 29894430, 29894430, 1159795426, 9069102398, 9069102398, 202847123212, 2237516341759, 2237516341759, 201635099759365, 1132157155708193, 6017397949439540, 17416293134812683, 496169890920484689, 1613261619087052703

Offset: 0

Seiichi Manyama, Aug 03 2019

nonn

			a(1) = (   6)_7 = 6,
a(2) = (  26)_7 = 20,
a(3) = ( 526)_7 = 265,
a(4) = (5526)_7 = 1980.

Cf. A309449.
Expansions of p-adic integers:
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));
A309451 (7-adic, 3^(1/5));
A309452 (7-adic, 4^(1/5));
A309453 (7-adic, 5^(1/5)).

PARI
```
{a(n) = truncate((6+O(7^n))^(1/5))}
```

a(0) = 0 and a(1) = 6, a(n) = a(n-1) + 4 * (a(n-1)^5 - 6) mod 7^n for n > 1.

A290796 Coefficients in 7-adic expansion of sqrt(-3).

Original entry on oeis.org

2, 5, 0, 6, 5, 2, 1, 1, 5, 6, 4, 4, 2, 1, 4, 2, 4, 5, 0, 3, 4, 0, 3, 5, 2, 0, 5, 3, 0, 5, 6, 2, 0, 0, 1, 2, 2, 6, 2, 5, 5, 1, 0, 3, 3, 4, 3, 2, 0, 5, 3, 0, 5, 5, 0, 0, 2, 5, 4, 4, 2, 3, 6, 5, 5, 2, 6, 0, 4, 6, 2, 1, 3, 3, 1, 3, 2, 0, 5, 1, 1, 5, 4, 1, 4, 1, 6, 4

Offset: 0

Seiichi Manyama, Aug 10 2017

nonn

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

Cf. A290797, A290803.

Vecrev(digits(lift(sqrt(-3+O(7^99))), 7))

A318960 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 1 (mod 4) case.

Original entry on oeis.org

1, 5, 5, 21, 53, 53, 181, 181, 181, 181, 181, 181, 181, 16565, 49333, 49333, 49333, 49333, 573621, 1622197, 1622197, 1622197, 10010805, 10010805, 10010805, 77119669, 211337397, 479772853, 479772853, 479772853, 2627256501, 6922223797, 15512158389, 15512158389

Offset: 2

Jianing Song, Sep 06 2018

nonn

a(n) is the unique number k in [1, 2^n] and congruent to 1 (mod 4) such that k^2 + 7 is divisible by 2^(n+1).

The 2-adic integers are very different from p-adic ones where p is an odd prime. For example, provided that there is at least one solution, the number of solutions to x^n = a over p-adic integers is gcd(n, p-1) for odd primes p and gcd(n, 2) for p = 2. For odd primes p, x^2 = a is solvable iff a is a quadratic residue modulo p, while for p = 2 it's solvable iff a == 1 (mod 8). If gcd(n, p-1) > 1 and gcd(a, p) = 1, then the solutions to x^n = a differ starting at the rightmost digit for odd primes p, while for p = 2 they differ starting at the next-to-rightmost digit. As a result, the formulas and the program here are different from those in other entries related to p-adic integers.

			The unique number k in [1, 4] and congruent to 1 modulo 4 such that k^2 + 7 is divisible by 8 is 1, so a(2) = 1.
a(2)^2 + 7 = 8 which is not divisible by 16, so a(3) = a(2) + 2^2 = 5.
a(3)^2 + 7 = 32 which is divisible by 32, so a(4) = a(3) = 5.
a(4)^2 + 7 = 32 which is divisible by 64, so a(5) = a(4) + 2^4 = 21.
a(5)^2 + 7 = 448 which is divisible by 128, so a(6) = a(5) + 2^5 = 53.
...

Jianing Song, Table of n, a(n) for n = 2..999 (offset corrected by Jianing Song)
G. P. Michon, Introduction to p-adic integers, Numericana.

Cf. A318962.
Expansions of p-adic integers:
this sequence, A318961 (2-adic, sqrt(-7));
A268924, A271222 (3-adic, sqrt(-2));
A268922, A269590 (5-adic, sqrt(-4));
A048898, A048899 (5-adic, sqrt(-1));
A290567 (5-adic, 2^(1/3));
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));
A286840, A286841 (13-adic, sqrt(-1));
A286877, A286878 (17-adic, sqrt(-1)).
Also expansions of 10-adic integers:
A007185, A010690 (nontrivial roots to x^2-x);
A216092, A216093, A224473, A224474 (nontrivial roots to x^3-x).

PARI
```
a(n) = truncate(-sqrt(-7+O(2^(n+1))))
```

a(2) = 1; for n >= 3, a(n) = a(n-1) if a(n-1)^2 + 7 is divisible by 2^(n+1), otherwise a(n-1) + 2^(n-1).
a(n) = 2^n - A318961(n).
a(n) = Sum_{i=0..n-1} A318962(i)*2^i.

Offset corrected by Jianing Song, Aug 28 2019

A318961 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 3 (mod 4) case.

Original entry on oeis.org

3, 3, 11, 11, 11, 75, 75, 331, 843, 1867, 3915, 8011, 16203, 16203, 16203, 81739, 212811, 474955, 474955, 474955, 2572107, 6766411, 6766411, 23543627, 57098059, 57098059, 57098059, 57098059, 593968971, 1667710795, 1667710795, 1667710795, 1667710795, 18847579979

Offset: 2

Jianing Song, Sep 06 2018

nonn

a(n) is the unique number k in [1, 2^n] and congruent to 3 (mod 4) such that k^2 + 7 is divisible by 2^(n+1).

The 2-adic integers are very different from p-adic ones where p is an odd prime. For example, provided that there is at least one solution, the number of solutions to x^n = a over p-adic integers is gcd(n, p-1) for odd primes p and gcd(n, 2) for p = 2. For odd primes p, x^2 = a is solvable iff a is a quadratic residue modulo p, while for p = 2 it's solvable iff a == 1 (mod 8). If gcd(n, p-1) > 1 and gcd(a, p) = 1, then the solutions to x^n = a differ starting at the rightmost digit for odd primes p, while for p = 2 they differ starting at the next-to-rightmost digit. As a result, the formulas and the program here are different from those in other entries related to p-adic integers.

			The unique number k in [1, 4] and congruent to 3 modulo 4 such that k^2 + 7 is divisible by 8 is 3, so a(2) = 3.
a(2)^2 + 7 = 16 which is divisible by 16, so a(3) = a(2) = 3.
a(3)^2 + 7 = 16 which is not divisible by 32, so a(4) = a(3) + 2^3 = 11.
a(4)^2 + 7 = 128 which is divisible by 64, so a(5) = a(4) = 11.
a(5)^2 + 7 = 128 which is divisible by 128, so a(6) = a(5) = 11.
...

Jianing Song, Table of n, a(n) for n = 2..999 (offset corrected by Jianing Song)
G. P. Michon, Introduction to p-adic integers, Numericana.

Cf. A318963.
Expansions of p-adic integers:
A318960, this sequence (2-adic, sqrt(-7));
A268924, A271222 (3-adic, sqrt(-2));
A268922, A269590 (5-adic, sqrt(-4));
A048898, A048899 (5-adic, sqrt(-1));
A290567 (5-adic, 2^(1/3));
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));
A286840, A286841 (13-adic, sqrt(-1));
A286877, A286878 (17-adic, sqrt(-1)).
Also expansions of 10-adic integers:
A007185, A010690 (nontrivial roots to x^2-x);
A216092, A216093, A224473, A224474 (nontrivial roots to x^3-x).

a(n) = if(n==2, 3, truncate(sqrt(-7+O(2^(n+1)))))

a(2) = 3; for n >= 3, a(n) = a(n-1) if a(n-1)^2 + 7 is divisible by 2^(n+1), otherwise a(n-1) + 2^(n-1).
a(n) = 2^n - A318960(n).
a(n) = Sum_{i=0..n-1} A318963(i)*2^i.

Offset corrected by Jianing Song, Aug 28 2019

cp's OEIS Frontend

A309450 The successive approximations up to 7^n for 7-adic integer 2^(1/5).

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A290804 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 5 mod 7 (except for the initial 0).

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

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A309452 The successive approximations up to 7^n for 7-adic integer 4^(1/5).

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A309453 The successive approximations up to 7^n for 7-adic integer 5^(1/5).

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A309454 The successive approximations up to 7^n for 7-adic integer 6^(1/5).

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A290796 Coefficients in 7-adic expansion of sqrt(-3).

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A318960 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 1 (mod 4) case.

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A318961 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 3 (mod 4) case.