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 1-7 of 7 results.

A320562 Irregular table read by rows: T(n,k) is the smallest m such that m^m == 2*k + 1 (mod 2^n), 0 <= k <= 2^(n-1) - 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 11, 5, 7, 9, 3, 13, 15, 1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31, 1, 27, 21, 55, 9, 19, 29, 47, 17, 11, 37, 39, 25, 3, 45, 31, 33, 59, 53, 23, 41, 51, 61, 15, 49, 43, 5, 7, 57, 35, 13, 63
Offset: 1

Views

Author

Jianing Song, Oct 15 2018

Keywords

Comments

The sequence {k^k mod 2^n} has period 2^n. The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
Note that the first 5 rows are the same as those in A320561, but after that they differ.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k. [Revised by Jianing Song, Nov 24 2018]
For n >= 3, T(n,k) = 2*k + 1 iff k is divisible by 2^floor((n-1)/2) or k = 2^(n-2) - 1 or k = 2^(n-1) - 1.
T(n,k) is the multiplicative inverse of A321906(n,k) modulo 2^n. - Jianing Song, Nov 24 2018

Examples

			Table starts
1,
1, 3,
1, 3, 5, 7,
1, 11, 5, 7, 9, 3, 13, 15,
1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31,
1, 27, 21, 55, 9, 19, 29, 47, 17, 11, 37, 39, 25, 3, 45, 31, 33, 59, 53, 23, 41, 51, 61, 15, 49, 43, 5, 7, 57, 35, 13, 63,
...

Links

Crossrefs

Cf. A007814.
{x^x} and its inverse: A320561 & this sequence.
{x^(-x)} and its inverse: A321901 & A321904.
{x^(1/x)} and its inverse: A321902 & A321905.
{x^(-1/x)} and its inverse: A321903 & A321906.

Programs

  • Mathematica
    Table[Block[{m = 1}, While[PowerMod[m, m, 2^n] != Mod[2 k + 1, 2^n], m++]; m], {n, 6}, {k, 0, 2^(n - 1) - 1}] // Flatten (* Michael De Vlieger, Oct 22 2018 *)
  • PARI
    T(n,k) = my(m=1); while(Mod(m, 2^n)^m!=2*k+1, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print);

Formula

For given n >= 2 and 0 <= k <= 2^(n-2) - 1, T(n,k) = T(n-1,k) if T(n-1,k)^T(n-1,k) == 2*k + 1 (mod 2^n), otherwise T(n-1,k) + 2^(n-1); for 2^(n-2) <= k <= 2^(n-1) - 1, T(n,k) = T(n,k-2^(n-2)) + 2^(n-1) if T(n,k) < 2^(n-1), otherwise T(n,k-2^(n-2)) - 2^(n-1).
T(n,k) = 2^n - A321904(n,2^(n-1)-1-k). - Jianing Song, Nov 24 2018

A321901 Irregular table read by rows: T(n,k) = (2*k+1)^(-(2*k+1)) mod 2^n, 0 <= k <= 2^(n-1) - 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 3, 13, 7, 9, 11, 5, 15, 1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31, 1, 19, 29, 7, 57, 27, 21, 15, 49, 35, 13, 23, 41, 43, 5, 31, 33, 51, 61, 39, 25, 59, 53, 47, 17, 3, 45, 55, 9, 11, 37, 63
Offset: 1

Views

Author

Jianing Song, Nov 21 2018

Keywords

Comments

The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k.
For n >= 3, T(n,k) = 2*k + 1 iff k == -1 (mod 2^floor((n-1)/2)) or k = 0 or k = 2^(n-2).
T(n,k) is the multiplicative inverse of A320561(n,k) modulo 2^n.

Examples

			Table starts
1,
1, 3,
1, 3, 5, 7,
1, 3, 13, 7, 9, 11, 5, 15,
1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31,
1, 19, 29, 7, 57, 27, 21, 15, 49, 35, 13, 23, 41, 43, 5, 31, 33, 51, 61, 39, 25, 59, 53, 47, 17, 3, 45, 55, 9, 11, 37, 63,
...

Crossrefs

Cf. A007814.
{x^x} and its inverse: A320561 & A320562.
{x^(-x)} and its inverse: this sequence & A321904.
{x^(1/x)} and its inverse: A321902 & A321905.
{x^(-1/x)} and its inverse: A321903 & A321906.

Programs

  • PARI
    T(n, k) = lift(Mod(2*k+1, 2^n)^(-(2*k+1)))
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Formula

T(n,k) = 2^n - A320561(n,2^(n-1)-1-k).

A321902 Irregular table read by rows: T(n,k) = (2*k+1)^(1/(2*k+1)) mod 2^n, 0 <= k <= 2^(n-1) - 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 11, 5, 7, 9, 3, 13, 15, 1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31, 1, 59, 21, 55, 9, 51, 29, 47, 17, 43, 37, 39, 25, 35, 45, 31, 33, 27, 53, 23, 41, 19, 61, 15, 49, 11, 5, 7, 57, 3, 13, 63
Offset: 1

Views

Author

Jianing Song, Nov 21 2018

Keywords

Comments

T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that x^(2*k+1) == 2*k + 1 (mod 2^n).
The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k.
For n >= 3, T(n,k) = 2*k + 1 iff k is divisible by 2^floor((n-1)/2) or k = 2^(n-2) - 1 or k = 2^(n-1) - 1.
T(n,k) is the multiplicative inverse of A321903(n,k) modulo 2^n.

Examples

			Table starts
1,
1, 3,
1, 3, 5, 7,
1, 11, 5, 7, 9, 3, 13, 15,
1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31,
1, 59, 21, 55, 9, 51, 29, 47, 17, 43, 37, 39, 25, 35, 45, 31, 33, 27, 53, 23, 41, 19, 61, 15, 49, 11, 5, 7, 57, 3, 13, 63,
...

Crossrefs

Cf. A007814.
{x^x} and its inverse: A320561 & A320562.
{x^(-x)} and its inverse: A321901 & A321904.
{x^(1/x)} and its inverse: this sequence & A321905.
{x^(-1/x)} and its inverse: A321903 & A321906.

Programs

  • PARI
    T(n, k) = my(m=1); while(Mod(m, 2^n)^(2*k+1)!=2*k+1, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Formula

T(n,k) = 2^n - A321903(n,2^(n-1)-1-k).

A321903 Irregular table read by rows: T(n,k) = (2*k+1)^(-1/(2*k+1)) mod 2^n, 0 <= k <= 2^(n-1) - 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 3, 13, 7, 9, 11, 5, 15, 1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31, 1, 51, 61, 7, 57, 59, 53, 15, 49, 3, 45, 23, 41, 11, 37, 31, 33, 19, 29, 39, 25, 27, 21, 47, 17, 35, 13, 55, 9, 43, 5, 63
Offset: 1

Views

Author

Jianing Song, Nov 21 2018

Keywords

Comments

T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that x^(-(2*k+1)) == 2*k + 1 (mod 2^n).
The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k.
For n >= 3, T(n,k) = 2*k + 1 iff k == -1 (mod 2^floor((n-1)/2)) or k = 0 or k = 2^(n-2).
T(n,k) is the multiplicative inverse of A321902(n,k) modulo 2^n.

Examples

			Table starts
1,
1, 3,
1, 3, 5, 7,
1, 3, 13, 7, 9, 11, 5, 15,
1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31,
1, 51, 61, 7, 57, 59, 53, 15, 49, 3, 45, 23, 41, 11, 37, 31, 33, 19, 29, 39, 25, 27, 21, 47, 17, 35, 13, 55, 9, 43, 5, 63,
...

Crossrefs

Cf. A007814.
{x^x} and its inverse: A320561 & A320562.
{x^(-x)} and its inverse: A321901 & A321904.
{x^(1/x)} and its inverse: A321902 & A321905.
{x^(-1/x)} and its inverse: this sequence & A321906.

Programs

  • PARI
    T(n, k) = my(m=1); while(Mod(m, 2^n)^(-(2*k+1))!=2*k+1, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Formula

T(n,k) = 2^n - A321902(n,2^(n-1)-1-k).

A321904 Irregular table read by rows: T(n,k) is the smallest m such that m^(-m) == 2*k + 1 (mod 2^n), 0 <= k <= 2^(n-1) - 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 3, 13, 7, 9, 11, 5, 15, 1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31, 1, 51, 29, 7, 57, 59, 21, 15, 49, 3, 13, 23, 41, 11, 5, 31, 33, 19, 61, 39, 25, 27, 53, 47, 17, 35, 45, 55, 9, 43, 37, 63
Offset: 1

Views

Author

Jianing Song, Nov 21 2018

Keywords

Comments

The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k.
For n >= 3, T(n,k) = 2*k + 1 iff k == -1 (mod 2^floor((n-1)/2)) or k = 0 or k = 2^(n-2).
T(n,k) is the multiplicative inverse of A321905(n,k) modulo 2^n.

Examples

			Table starts
1,
1, 3,
1, 3, 5, 7,
1, 3, 13, 7, 9, 11, 5, 15,
1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31,
1, 51, 29, 7, 57, 59, 21, 15, 49, 3, 13, 23, 41, 11, 5, 31, 33, 19, 61, 39, 25, 27, 53, 47, 17, 35, 45, 55, 9, 43, 37, 63,
...

Crossrefs

Cf. A007814.
{x^x} and its inverse: A320561 & A320562.
{x^(-x)} and its inverse: A321901 & this sequence.
{x^(1/x)} and its inverse: A321902 & A321905.
{x^(-1/x)} and its inverse: A321903 & A321906.

Programs

  • PARI
    T(n, k) = my(m=1); while(Mod(m, 2^n)^(-m)!=2*k+1, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Formula

T(n,k) = 2^n - A320562(n,2^(n-1)-1-k).

A321905 Irregular table read by rows: T(n,k) is the smallest m such that m^(1/m) == 2*k + 1 (mod 2^n), 0 <= k <= 2^(n-1) - 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 11, 5, 7, 9, 3, 13, 15, 1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31, 1, 59, 53, 55, 9, 51, 61, 47, 17, 43, 5, 39, 25, 35, 13, 31, 33, 27, 21, 23, 41, 19, 29, 15, 49, 11, 37, 7, 57, 3, 45, 63
Offset: 1

Views

Author

Jianing Song, Nov 21 2018

Keywords

Comments

T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that (2*k+1)^m == m (mod 2^n).
The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k.
For n >= 3, T(n,k) = 2*k + 1 iff k is divisible by 2^floor((n-1)/2) or k = 2^(n-2) - 1 or k = 2^(n-1) - 1.
T(n,k) is the multiplicative inverse of A321904(n,k) modulo 2^n.

Examples

			Table starts
1,
1, 3,
1, 3, 5, 7,
1, 11, 5, 7, 9, 3, 13, 15,
1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31,
1, 59, 53, 55, 9, 51, 61, 47, 17, 43, 5, 39, 25, 35, 13, 31, 33, 27, 21, 23, 41, 19, 29, 15, 49, 11, 37, 7, 57, 3, 45, 63,
...

Crossrefs

Cf. A007814.
{x^x} and its inverse: A320561 & A320562.
{x^(-x)} and its inverse: A321901 & A321904.
{x^(1/x)} and its inverse: A321902 & this sequence.
{x^(-1/x)} and its inverse: A321903 & A321906.

Programs

  • PARI
    T(n, k) = my(m=1); while(Mod(2*k+1, 2^n)^m!=m, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Formula

T(n,k) = 2^n - A321906(n,2^(n-1)-1-k).

A321906 Irregular table read by rows: T(n,k) is the smallest m such that m^(-1/m) == 2*k + 1 (mod 2^n), 0 <= k <= 2^(n-1) - 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 3, 13, 7, 9, 11, 5, 15, 1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31, 1, 19, 61, 7, 57, 27, 53, 15, 49, 35, 45, 23, 41, 43, 37, 31, 33, 51, 29, 39, 25, 59, 21, 47, 17, 3, 13, 55, 9, 11, 5, 63
Offset: 1

Views

Author

Jianing Song, Nov 21 2018

Keywords

Comments

T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that (2*k+1)^(-m) == m (mod 2^n).
The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k.
For n >= 3, T(n,k) = 2*k + 1 iff k == -1 (mod 2^floor((n-1)/2)) or k = 0 or k = 2^(n-2).
T(n,k) is the multiplicative inverse of A320562(n,k) modulo 2^n.

Examples

			Table starts
1,
1, 3,
1, 3, 5, 7,
1, 3, 13, 7, 9, 11, 5, 15,
1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31,
1, 19, 61, 7, 57, 27, 53, 15, 49, 35, 45, 23, 41, 43, 37, 31, 33, 51, 29, 39, 25, 59, 21, 47, 17, 3, 13, 55, 9, 11, 5, 63,
...

Crossrefs

Cf. A007814.
{x^x} and its inverse: A320561 & A320562.
{x^(-x)} and its inverse: A321901 & A321904.
{x^(1/x)} and its inverse: A321902 & A321905.
{x^(-1/x)} and its inverse: A321903 & this sequence.

Programs

  • PARI
    T(n, k) = my(m=1); while(Mod(2*k+1, 2^n)^(-m)!=m, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Formula

T(n,k) = 2^n - A321905(n,2^(n-1)-1-k).
Showing 1-7 of 7 results.

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