cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A309449 Coefficients in 7-adic expansion of 6^(1/5).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Aug 03 2019

Keywords

Links

Crossrefs

Cf. A309454.
Digits of p-adic integers:
A309445 (7-adic, 2^(1/5));
A309446 (7-adic, 3^(1/5));
A309447 (7-adic, 4^(1/5));
A309448 (7-adic, 5^(1/5)).

Programs

  • Maple
    op([1,3],padic:-rootp(x^5-6, 7, 101)); # Robert Israel, Aug 04 2019
  • PARI
    Vecrev(digits(truncate((6+O(7^100))^(1/5)), 7))

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Aug 03 2019

Keywords

Examples

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

Links

Crossrefs

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)).

Programs

  • Maple
    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))}

Formula

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

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

Views

Author

Seiichi Manyama, Aug 03 2019

Keywords

Examples

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

Crossrefs

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)).

Programs

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

Formula

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

Views

Author

Seiichi Manyama, Aug 03 2019

Keywords

Examples

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

Crossrefs

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)).

Programs

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

Formula

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

Views

Author

Seiichi Manyama, Aug 03 2019

Keywords

Examples

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

Crossrefs

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)).

Programs

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

Formula

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

A322157 The successive approximations up to 5^n for 5-adic integer 7^(1/5).

Original entry on oeis.org

0, 2, 22, 47, 422, 1047, 13547, 29172, 341672, 732297, 732297, 30029172, 127685422, 860107297, 4522216672, 10625732297, 41143310422, 498906982297, 1261846435422, 5076543701047, 62297002685422, 348399297607297, 1778910772216672, 8931468145263547, 20852397100341672
Offset: 0

Views

Author

Jianing Song, Aug 28 2019

Keywords

Comments

For n > 0, a(n) is the unique number k in [1, 5^n] such that k^5 - 7 is divisible by 5^(n+1).
For k not divisible by 5, k is a fifth power in 5-adic field if and only if k == 1, 7, 18, 24 (mod 25). If k is a fifth power in 5-adic field, then k has exactly one fifth root.

Examples

			For n = 5, we have 1047^5 - 7 = 5^6 * 80521782896, and that 1047 is the unique number k in [1, 5^5] such that k^5 - 7 is divisible by 5^6, so a(5) = 1047.

Links

Crossrefs

For the digits of this number see A322169.
For fifth roots in 7-adic field, see A309450, A309451, A309452, A309453, A309454.

Programs

  • PARI
    a(n) = if(n, lift(sqrtn(7+O(5^(n+1)), 5)), 0)

A325892 The successive approximations up to 2^n for the 2-adic integer 3^(1/5).

Original entry on oeis.org

0, 1, 3, 3, 3, 19, 19, 83, 211, 211, 211, 211, 2259, 6355, 14547, 30931, 63699, 129235, 129235, 129235, 129235, 129235, 129235, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 2151807187, 6446774483, 6446774483, 6446774483
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

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

Examples

			For n = 2, the unique solution to x^5 == 3 (mod 4) in the range [0, 3] is x = 3, so a(2) = 3.
a(2)^5 - 3 = 240 which is divisible by 8, so a(3) = a(2) = 3;
a(3)^5 - 3 = 240 which is divisible by 16, so a(4) = a(3) = 3;
a(4)^5 - 3 = 240 which is not divisible by 32, so a(5) = a(4) + 16 = 19;
a(5)^5 - 3 = 2476096 which is divisible by 64, so a(6) = a(5) = 19.

Links

Crossrefs

For the digits of 3^(1/5), see A325896.
Approximations of p-adic fifth-power roots:
this sequence (2-adic, 3^(1/5));
A325893 (2-adic, 5^(1/5));
A325894 (2-adic, 7^(1/5));
A325895 (2-adic, 9^(1/5));
A322157 (5-adic, 7^(1/5));
A309450 (7-adic, 2^(1/5));
A309451 (7-adic, 3^(1/5));
A309452 (7-adic, 4^(1/5));
A309453 (7-adic, 5^(1/5));
A309454 (7-adic, 6^(1/5)).

Programs

  • PARI
    a(n) = lift(sqrtn(3+O(2^n), 5))

Formula

For n > 0, a(n) = a(n-1) if a(n-1)^5 - 3 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).

A325893 The successive approximations up to 2^n for 2-adic integer 5^(1/5).

Original entry on oeis.org

0, 1, 1, 5, 5, 21, 21, 21, 149, 149, 149, 1173, 1173, 1173, 9365, 9365, 42133, 107669, 238741, 500885, 1025173, 1025173, 1025173, 1025173, 1025173, 1025173, 34579605, 34579605, 34579605, 34579605, 571450517, 1645192341, 1645192341, 1645192341, 10235126933, 10235126933
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

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

Examples

			For n = 2, the unique solution to x^5 == 5 (mod 4) in the range [0, 3] is x = 1, so a(2) = 1.
a(2)^5 - 5 = -4 which is not divisible by 8, so a(3) = a(2) + 4 = 5;
a(3)^5 - 5 = 3120 which is divisible by 16, so a(4) = a(3) = 5;
a(4)^5 - 5 = 3120 which is not divisible by 32, so a(5) = a(4) + 16 = 21;
a(5)^5 - 5 = 4084096 which is divisible by 64, so a(6) = a(5) = 21.

Links

Crossrefs

For the digits of 5^(1/5), see A325897.
Approximations of p-adic fifth-power roots:
A325892 (2-adic, 3^(1/5));
this sequence (2-adic, 5^(1/5));
A325894 (2-adic, 7^(1/5));
A325895 (2-adic, 9^(1/5));
A322157 (5-adic, 7^(1/5));
A309450 (7-adic, 2^(1/5));
A309451 (7-adic, 3^(1/5));
A309452 (7-adic, 4^(1/5));
A309453 (7-adic, 5^(1/5));
A309454 (7-adic, 6^(1/5)).

Programs

  • PARI
    a(n) = lift(sqrtn(5+O(2^n), 5))

Formula

For n > 0, a(n) = a(n-1) if a(n-1)^5 - 5 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).

A325894 The successive approximations up to 2^n for the 2-adic integer 7^(1/5).

Original entry on oeis.org

0, 1, 3, 7, 7, 7, 39, 103, 231, 231, 743, 1767, 1767, 1767, 1767, 18151, 18151, 18151, 18151, 18151, 542439, 1591015, 3688167, 7882471, 16271079, 33048295, 66602727, 133711591, 267929319, 267929319, 804800231, 804800231, 804800231, 804800231, 9394734823, 26574604007
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

a(n) is the unique solution to x^5 == 7 (mod 2^n) in the range [0, 2^n - 1].

Examples

			For n = 2, the unique solution to x^5 == 7 (mod 4) in the range [0, 3] is x = 3, so a(2) = 3.
a(2)^5 - 7 = 236 which is not divisible by 8, so a(3) = a(2) + 4 = 7;
a(3)^5 - 7 = 16800 which is divisible by 16, so a(4) = a(3) = 7;
a(4)^5 - 7 = 16800 which is divisible by 32, so a(5) = a(4) = 7;
a(5)^5 - 7 = 16800 which is not divisible by 64, so a(6) = a(5) + 32 = 39.

Links

Crossrefs

For the digits of 7^(1/5), see A325898.
Approximations of p-adic fifth-power roots:
A325892 (2-adic, 3^(1/5));
A325893 (2-adic, 5^(1/5));
this sequence (2-adic, 7^(1/5));
A325895 (2-adic, 9^(1/5));
A322157 (5-adic, 7^(1/5));
A309450 (7-adic, 2^(1/5));
A309451 (7-adic, 3^(1/5));
A309452 (7-adic, 4^(1/5));
A309453 (7-adic, 5^(1/5));
A309454 (7-adic, 6^(1/5)).

Programs

  • PARI
    a(n) = lift(sqrtn(7+O(2^n), 5))

Formula

For n > 0, a(n) = a(n-1) if a(n-1)^5 - 7 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).

A325895 The successive approximations up to 2^n for the 2-adic integer 9^(1/5).

Original entry on oeis.org

0, 1, 1, 1, 9, 9, 41, 105, 233, 489, 489, 1513, 3561, 7657, 15849, 32233, 32233, 97769, 228841, 490985, 1015273, 2063849, 4161001, 8355305, 16743913, 16743913, 16743913, 83852777, 218070505, 218070505, 218070505, 1291812329, 1291812329, 5586779625, 5586779625, 5586779625
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

a(n) is the unique solution to x^5 == 9 (mod 2^n) in the range [0, 2^n - 1].

Examples

			For n = 2, the unique solution to x^5 == 9 (mod 4) in the range [0, 3] is x = 1, so a(2) = 1.
a(2)^5 - 9 = -8 which is divisible by 8, so a(3) = a(2) = 1;
a(3)^5 - 9 = -8 which is not divisible by 16, so a(4) = a(3) + 8 = 9;
a(4)^5 - 9 = 59040 which is divisible by 32, so a(5) = a(4) = 9;
a(5)^5 - 9 = 59040 which is not divisible by 64, so a(6) = a(5) + 32 = 41.

Links

Crossrefs

For the digits of 9^(1/5), see A325899.
Approximations of p-adic fifth-power roots:
A325892 (2-adic, 3^(1/5));
A325893 (2-adic, 5^(1/5));
A325894 (2-adic, 7^(1/5));
this sequence (2-adic, 9^(1/5));
A322157 (5-adic, 7^(1/5));
A309450 (7-adic, 2^(1/5));
A309451 (7-adic, 3^(1/5));
A309452 (7-adic, 4^(1/5));
A309453 (7-adic, 5^(1/5));
A309454 (7-adic, 6^(1/5)).

Programs

  • PARI
    a(n) = lift(sqrtn(9+O(2^n), 5))

Formula

For n > 0, a(n) = a(n-1) if a(n-1)^5 - 9 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).
Showing 1-10 of 10 results.

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