A048898 -id:A048898 - cp's OEIS frontend

A212152 Digits of one of the three 7-adic integers (-1)^(1/3).

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

Offset: 0

Wolfdieter Lang, May 02 2012

See A210852 for comments and an approximation to this 7-adic number, called there u.  See also A048898 for references on p-adic numbers.
a(n), n>=1, is the (unique) solution of the linear congruence 3 * b(n)^2 * a(n) + c(n) == 0 (mod 7), with  b(n):=A210852(n) and c(n):=A210853(n). a(0) = 3, one of the three solutions of x^3+1 == 0 (mod 7).
Since b(n) == 3 (mod 7), a(n) == c(n) (mod 7) for n>0. - Álvar Ibeas, Feb 20 2017
With a(0) = 2, this is the digits of one of the three cube root of 1, the one that is congruent to 2 modulo 7. - Jianing Song, Aug 26 2022

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A210852 (approximations of (-1)^(1/3)), A212155 (digits of another cube root of -1), 6*A000012 (digits of -1).

Cf. A210850, A210851 (digits of the 5-adic integers sqrt(-1)); A319297, A319305, A319555 (digits of the 7-adic integers 6^(1/3)).

op([1,1,3],select(t -> padic:-ratvaluep(t,1)=3, [padic:-rootp(x^3+1,7,100)])); # Robert Israel, Mar 27 2018

Join[{3}, MapIndexed[#/7^#2[[1]] &, Differences[FoldList[PowerMod[#, 7, 7^#2] &, 3, Range[2, 100]]]]] (* Paolo Xausa, Jan 14 2025 *)

a(n) = (b(n+1) - b(n))/7^n, n>=1, with b(n):=A210852(n), defined by a recurrence given there. One also finds a Maple program for b(n) there. a(0)=3.

A212155 Digits of one of the three 7-adic integers (-1)^(1/3).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, May 02 2012

See A210853 for comments and an approximation to this 7-adic number, called there v.  See also A048898 for references on p-adic numbers.
a(n), n>=1, is the (unique) solution of the linear congruence 3 * b(n)^2 * a(n) + c(n) == 0 (mod 7), with  b(n):=A212153(n) and c(n):=A212154(n). a(0) = 5, one of the three solutions of X^3+1 == 0 (mod 7).
Since b(n) == 5 (mod 7), a(n) == 4 * c(n) (mod 7) for n>0. - Álvar Ibeas, Feb 20 2017
With a(0) = 4, this is the digits of one of the three cube root of 1, the one that is congruent to 4 modulo 7. - Jianing Song, Aug 26 2022

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A212153 (approximations of (-1)^(1/3)), A212152 (digits of another cube root of -1), 6*A000012 (digits of -1).

Cf. A210850, A210851 (digits of the 5-adic integers sqrt(-1)); A319297, A319305, A319555 (digits of the 7-adic integers 6^(1/3)).

Join[{5}, MapIndexed[#/7^#2[[1]] &, Differences[FoldList[PowerMod[#, 7, 7^#2] &, 5, Range[2, 100]]]]] (* Paolo Xausa, Jan 14 2025 *)

a(n) = (b(n+1) - b(n))/7^n, n>=1, with b(n):=A212153(n), defined by a recurrence given there. One also finds there a Maple program for b(n). a(0)=5.

a(n) = 6 - A212152(n), for n>0. - Álvar Ibeas, Feb 21 2017

A210849 a(n) = (A048899(n)^2 + 1)/5^n, n >= 0.

Original entry on oeis.org

1, 2, 13, 37, 314, 365, 73, 13369, 31226, 1432954, 1346393, 10982633, 59784881, 986508685, 197301737, 12342639754, 16335212753, 165277755905, 33055551181, 12781804411945, 2556360882389, 25830314642530

Offset: 0

Wolfdieter Lang, Apr 28 2012

a(n) is an integer (nonnegative) because b(n):=A048899(n) satisfies b(n)^2 + 1 == 0 (mod 5^n), n>=0. The solution of this congruence for n>=1, which satisfies also b(n) == 3 (mod 5), is b(n) = 3^(5^(n-1)) (mod 5^n), but this is inconvenient for computing b(n) for large n. Instead one can use the b(n) recurrence which follows immediately, and this is given in the formula field below. To prove that the given b(n) formula solves the first congruence one can analyze the binomial expansion of (10 - 1)^(5^(n-1)) + 1 and show that it is 0 (mod 5^n) term by term. The second congruence reduces to b(n) == 3^(5^(n-1)) (mod 5) which follows for n>=1 by induction. Because b(n) = 5^n - A048898(n) one could also use the result A048898(n) == 2 (mod 5) once this is proved.

See also the comment on A210848 on two relevant theorems.

			a(0) = 1/1 = 1.
a(3) = (68^2 + 1)/5^3 = 37  (b(3) = 18^5 (mod 5^3) = 68).

Cf. A048899, A048898, A210848 (companion sequence).

b:=proc(n) option remember: if n=0 then 0 elif n=1 then 3
else modp(b(n-1)^5,5^n) fi end proc:
[seq((b(n)^2+1)/5^n,n=0..29)];

b[n_] := b[n] = Which[n == 0, 0, n == 1, 3, True, Mod[b[n-1]^5, 5^n]]; Table[(b[n]^2+1)/5^n, {n, 0, 29}] (* Jean-François Alcover, Mar 05 2014, after Maple *)

a(n) = (b(n)^2 + 1)/5^n, n>=0, with b(n) = A048899(n) given by the recurrence b(n) = b(n-1)^5 (mod 5^n), n>=2, b(0):=0, b(1)=3 (this is the analog of the Mathematica Program by Jean-François Alcover for A048898).

a(n) - A210848(n) = A048899(n) - A048898(n) (== 1 mod 5 if n>0). - Álvar Ibeas, Feb 21 2017

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

A324027 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 2 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 2, 12, 37, 162, 1412, 10787, 42037, 354537, 1526412, 3479537, 3479537, 3479537, 247620162, 3909729537, 10013245162, 101565979537, 711917542037, 2237796448287, 13681888245162, 51828860901412, 337931155823287, 1291605472229537, 10828348636292037, 58512064456604537

Offset: 0

Jianing Song, Sep 07 2019

nonn

For n > 0, a(n) is the unique solution to x^2 == -6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 2 modulo 5.

A324028 is the approximation (congruent to 3 mod 5) of another square root of 6 over the 5-adic field.

			12^2 = 144 = 6*5^2 - 6;
37^2 = 1369 = 11*5^3 - 6;
162^2 = 26244 = 42*5^4 - 6.

Wikipedia, p-adic number

Cf. A048898, A048899, A324029, A324030.
Approximations of 5-adic square roots:
this sequence, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

PARI
```
a(n) = truncate(sqrt(-6+O(5^n)))
```

For n > 0, a(n) = 5^n - A324028(n).

a(n) = A048898(n)*A324023(n) mod 5^n = A048899(n)*A324024(n) mod 5^n.

A324028 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 3 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 3, 13, 88, 463, 1713, 4838, 36088, 36088, 426713, 6286088, 45348588, 240661088, 973082963, 2193786088, 20504332963, 51021911088, 51021911088, 1576900817338, 5391598082963, 43538570739213, 138906002379838, 1092580318786088, 1092580318786088, 1092580318786088

Offset: 0

Jianing Song, Sep 07 2019

nonn

For n > 0, a(n) is the unique solution to x^2 == -6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 3 modulo 5.

A324027 is the approximation (congruent to 3 mod 5) of another square root of -6 over the 5-adic field.

			13^2 = 169 = 7*5^2 - 6;
88^2 = 7744 = 62*5^3 - 6;
463^2 = 214369 = 343*5^4 - 6.

Wikipedia, p-adic number

Cf. A048898, A048899, A324029, A324030.
Approximations of 5-adic square roots:
A324027, sequence (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

PARI
```
a(n) = truncate(-sqrt(-6+O(5^n)))
```

For n > 0, a(n) = 5^n - A324027(n).

a(n) = A048898(n)*A324024(n) mod 5^n = A048899(n)*A324023(n) mod 5^n.

A034935 Successive approximations to 5-adic integer sqrt(-1).

Original entry on oeis.org

0, 2, 7, 57, 182, 2057, 14557, 45807, 280182, 6139557, 25670807, 123327057, 5006139557, 11109655182, 102662389557, 407838170807, 3459595983307, 79753541295807, 365855836217682, 2273204469030182, 49956920289342682

Offset: 0

N. J. A. Sloane

This is the root congruent to 2 mod 5.

J. H. Conway, The Sensual Quadratic Form, p. 118.
K. Mahler, Introduction to p-Adic Numbers and Their Functions, Cambridge, 1973, p. 35.

Seiichi Manyama, Table of n, a(n) for n = 0..1121

Cf. A034939, A048898, A048899.

Table[ PowerMod[2, 5^n, 5^n], {n, 0, 24}] // Union (* Jean-François Alcover, Dec 03 2012, from formula given by Joe K. Crump *)

PARI
```
sqrt(-1+O(5^40))
```

{a(n) = local(k, x, y); for(i = 0, n, until( x != (y = truncate( sqrt( -1 + O(5^(k++))))), x = y));x} /* Michael Somos, Mar 03 2008 */

Successive values of 2^(5^x) mod 5^x. - Joe K. Crump (joecr(AT)carolina.rr.com), Jan 20 2001

A327303 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 4 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 4, 4, 79, 79, 79, 3204, 18829, 331329, 1112579, 1112579, 20643829, 118300079, 850721954, 3292128204, 27706190704, 149776503204, 302364393829, 1065303846954, 8694698378204, 46841671034454, 332943965956329, 332943965956329, 5101315547987579, 28943173458143829

Offset: 0

Jianing Song, Sep 16 2019

nonn

a(n) is the unique number k in [1, 5^n] and congruent to 4 mod 5 such that k^2 + 9 is divisible by 5^n.

			The unique number k in {4, 9, 14, 19, 24} such that k^2 + 9 is divisible by 25 is k = 4, so a(2) = 4.
The unique number k in {4, 29, 54, 79, 104} such that k^2 + 9 is divisible by 125 is k = 79, so a(3) = 46.
The unique number k in {79, 204, 329, 454, 579} such that k^2 + 9 is divisible by 625 is k = 79, so a(4) = 79.

Robert Israel, Table of n, a(n) for n = 0..1424
G. P. Michon, Introduction to p-adic integers, Numericana.

For the digits of sqrt(-9) see A327304 and A327305.
Approximations of 5-adic square roots:
A327302, this sequence (sqrt(-9));
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

R:= [padic:-rootp(x^2+9,5,101)]:
R:= op(select(t -> padic:-ratvaluep(t,1)=4, R)):
seq(padic:-ratvaluep(R,n),n=0..100); # Robert Israel, Jan 16 2023

PARI
```
a(n) = truncate(-sqrt(-9+O(5^n)))
```

a(1) = 4; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.

For n > 0, a(n) = 5^n - A327302(n).

A327302 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 1 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 1, 21, 46, 546, 3046, 12421, 59296, 59296, 840546, 8653046, 28184296, 125840546, 369981171, 2811387421, 2811387421, 2811387421, 460575059296, 2749393418671, 10378787949921, 48525760606171, 143893192246796, 2051241825059296, 6819613407090546, 30661471317246796

Offset: 0

Jianing Song, Sep 16 2019

nonn

a(n) is the unique number k in [1, 5^n] and congruent to 1 mod 5 such that k^2 + 9 is divisible by 5^n.

			The unique number k in {1, 6, 11, 16, 21} such that k^2 + 9 is divisible by 25 is k = 21, so a(2) = 21.
The unique number k in {21, 46, 71, 96, 121} such that k^2 + 9 is divisible by 125 is k = 46, so a(3) = 46.
The unique number k in {46, 171, 296, 421, 546} such that k^2 + 9 is divisible by 625 is k = 546, so a(4) = 546.

G. P. Michon, Introduction to p-adic integers, Numericana.

For the digits of sqrt(-9) see A327304 and A327305.
Approximations of 5-adic square roots:
this sequence, A327303 (sqrt(-9));
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

PARI
```
a(n) = truncate(sqrt(-9+O(5^n)))
```

a(1) = 1; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.

For n > 0, a(n) = 5^n - A327303(n).

cp's OEIS Frontend

A212152 Digits of one of the three 7-adic integers (-1)^(1/3).

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A212155 Digits of one of the three 7-adic integers (-1)^(1/3).

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A210849 a(n) = (A048899(n)^2 + 1)/5^n, n >= 0.

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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.

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A324027 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 2 (mod 5) case (except for n = 0).

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A324028 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 3 (mod 5) case (except for n = 0).

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A034935 Successive approximations to 5-adic integer sqrt(-1).