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 1-5 of 5 results.

A370067 Square array read by ascending antidiagonals: T(n,k) is the size of the group Q_p*/(Q_p*)^k, where p = prime(n), and Q_p is the field of p-adic numbers.

Original entry on oeis.org

1, 1, 8, 1, 4, 3, 1, 4, 9, 32, 1, 4, 3, 8, 5, 1, 4, 9, 16, 5, 24, 1, 4, 3, 8, 25, 36, 7, 1, 4, 9, 8, 5, 12, 7, 128, 1, 4, 3, 16, 25, 36, 7, 16, 9, 1, 4, 9, 16, 5, 12, 49, 32, 81, 40, 1, 4, 3, 8, 5, 36, 7, 16, 9, 20, 11, 1, 4, 3, 8, 5, 12, 7, 16, 27, 100, 11, 96, 1, 4, 9, 16, 5, 36, 7, 32, 9, 20, 11, 72, 13
Offset: 1

Views

Author

Jianing Song, Apr 30 2024

Keywords

Comments

We have Q_p* = p^Z X Z_p*, so Q_p*/(Q_p*)^k = (p^Z/p^(kZ)) X (Z_p*/(Z_p*)^k). Note that p^Z/p^(kZ) is a cyclic group of order k. For the group structure of (Z_p*/(Z_p*)^k), see A370050.
Each row is multiplicative.

Examples

			Table reads
  1, 8, 3, 32, 5, 24, 7, 128, 9, 40
  1, 4, 9, 8, 5, 36, 7, 16, 81, 20
  1, 4, 3, 16, 25, 12, 7, 32, 9, 100
  1, 4, 9, 8, 5, 36, 49, 16, 27, 20
  1, 4, 3, 8, 25, 12, 7, 16, 9, 100
  1, 4, 9, 16, 5, 36, 7, 32, 27, 20
  1, 4, 3, 16, 5, 12, 7, 64, 9, 20
  1, 4, 9, 8, 5, 36, 7, 16, 81, 20
  1, 4, 3, 8, 5, 12, 7, 16, 9, 20
  1, 4, 3, 16, 5, 12, 49, 32, 9, 20

Links

Crossrefs

Cf. A370050. Row 1-4: A370564, A370565, A370566, A370567.

Programs

  • PARI
    T(n, k) = my(p = prime(n), e = valuation(k, p)); k * p^e*gcd(p-1, k/p^e) * if(p==2 && e>=1, 2, 1)

Formula

T(n,k) = k * A370050(n,k).
Write k = p^e * k' with k' not being divisible by p, and p = prime(n). If p is odd, then T(n,k) = k * p^e * gcd(p-1,k'). If p = 2 and k is odd, then T(n,k) = k. If p = 2 and k is even, then T(n,k) = k * 2^(e+1).

A370566 Size of the group Q_5*/(Q_5*)^n, where Q_5 is the field of 5-adic numbers.

Original entry on oeis.org

1, 4, 3, 16, 25, 12, 7, 32, 9, 100, 11, 48, 13, 28, 75, 64, 17, 36, 19, 400, 21, 44, 23, 96, 625, 52, 27, 112, 29, 300, 31, 128, 33, 68, 175, 144, 37, 76, 39, 800, 41, 84, 43, 176, 225, 92, 47, 192, 49, 2500, 51, 208, 53, 108, 275, 224, 57, 116, 59, 1200, 61, 124, 63, 256
Offset: 1

Views

Author

Jianing Song, Apr 30 2024

Keywords

Comments

We have Q_5* = 5^Z X Z_5*, so Q_5*/(Q_5*)^k = (5^Z/5^(kZ)) X (Z_p*/(Z_5*)^k). Note that 5^Z/5^(kZ) is a cyclic group of order k. For the group structure of (Z_5*/(Z_5*)^k), see A370050.

Links

Crossrefs

Row 3 of A370067. Cf. A001620, A370050, A370564, A370565, A370567.
Cf. A370181.

Programs

  • Mathematica
    a[n_] := Module[{e2 = IntegerExponent[n, 2], e5 = IntegerExponent[n, 5]}, 2^Min[e2, 2] * 5^e5 * n]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)
  • PARI
    a(n, {p=5}) = my(e = valuation(n, p)); n * p^e*gcd(p-1, n/p^e)

Formula

Write n = 5^e * n' with k' not being divisible by 5, then a(n) = n * 5^e * gcd(4,n').
Multiplicative with a(5^e) = 5^(2*e), a(2) = 4, a(2^e) = 2^(e+2) for e >= 2 and a(p^e) = p^e for primes p != 2, 5.
a(n) = n * A370181(n).
From Amiram Eldar, May 20 2024: (Start)
Dirichlet g.f.: ((1 + 1/2^(s-1) + 1/2^(2*s-3)) * (1 - 1/5^(s-1))/(1 - 1/5^(s-2))) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (4*n^2/(5*log(5))) * (log(n) + gamma - 1/2 + 3*log(5/2)/4), where gamma is Euler's constant (A001620). (End)

A370567 Size of the group Q_7*/(Q_7*)^n, where Q_7 is the field of 7-adic numbers.

Original entry on oeis.org

1, 4, 9, 8, 5, 36, 49, 16, 27, 20, 11, 72, 13, 196, 45, 32, 17, 108, 19, 40, 441, 44, 23, 144, 25, 52, 81, 392, 29, 180, 31, 64, 99, 68, 245, 216, 37, 76, 117, 80, 41, 1764, 43, 88, 135, 92, 47, 288, 2401, 100, 153, 104, 53, 324, 55, 784, 171, 116, 59, 360, 61, 124, 1323, 128
Offset: 1

Views

Author

Jianing Song, Apr 30 2024

Keywords

Comments

We have Q_7* = 7^Z X Z_7*, so Q_7*/(Q_7*)^k = (7^Z/7^(kZ)) X (Z_p*/(Z_7*)^k). Note that 7^Z/7^(kZ) is a cyclic group of order k. For the group structure of (Z_7*/(Z_7*)^k), see A370050.

Links

Crossrefs

Row 4 of A370067. Cf. A001620, A370050, A370564, A370565, A370566.
Cf. A370182.

Programs

  • Mathematica
    a[n_] := Module[{e2 = IntegerExponent[n, 2], e3 = IntegerExponent[n, 3], e7 = IntegerExponent[n, 7]}, 2^Min[e2, 1] * 3^Min[e3, 1] * 7^e7 * n]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)
  • PARI
    a(n, {p=7}) = my(e = valuation(n, p)); n * p^e*gcd(p-1, n/p^e)

Formula

Write n = 7^e * n' with k' not being divisible by 7, then a(n) = n * 7^e * gcd(6,n').
Multiplicative with a(7^e) = 7^(2*e), a(2^e) = 2^(e+1), a(3^e) = 3^(e+1) and a(p^e) = p^e for primes p != 2, 3, 7.
a(n) = n * A370182(n).
From Amiram Eldar, May 20 2024: (Start)
Dirichlet g.f.: ((1 + 1/2^(s-1)) * (1 + 2/3^(s-1)) * (1 - 1/7^(s-1))/(1 - 1/7^(s-2))) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (15*n^2/(14*log(7))) * (log(n) + gamma - 1/2 + 2*log(7)/3 - 2*log(3)/5 - log(2)/3), where gamma is Euler's constant (A001620). (End)

A370180 Size of the group Z_3*/(Z_3*)^n, where Z_3 is the ring of 3-adic integers.

Original entry on oeis.org

1, 2, 3, 2, 1, 6, 1, 2, 9, 2, 1, 6, 1, 2, 3, 2, 1, 18, 1, 2, 3, 2, 1, 6, 1, 2, 27, 2, 1, 6, 1, 2, 3, 2, 1, 18, 1, 2, 3, 2, 1, 6, 1, 2, 9, 2, 1, 6, 1, 2, 3, 2, 1, 54, 1, 2, 3, 2, 1, 6, 1, 2, 9, 2, 1, 6, 1, 2, 3, 2, 1, 18, 1, 2, 3, 2, 1, 6, 1, 2, 81, 2, 1, 6, 1, 2, 3, 2, 1, 18
Offset: 1

Views

Author

Jianing Song, Apr 30 2024

Keywords

Comments

We have that Z_3*/(Z_3*)^n is the inverse limit of (Z/3^iZ)*/((Z/3^iZ)*)^n as i tends to infinity. Write n = 3^e * n' with n' not being divisible by 3, then the group is cyclic of order 3^e * gcd(2,n'). See A370050.

Examples

			We have Z_3*/(Z_3*)^3 = Z_3* / ((1+9Z_3) U (8+9Z_3)) = (Z/9Z)*/((1+9Z) U (8+9Z)) = C_3, so a(3) = 3.
We have Z_3*/(Z_3*)^6 = Z_3* / (1+9Z_3) = (Z/9Z)*/(1+9Z) = C_6, so a(6) = 6.

Links

Crossrefs

Row 2 of A370050. Cf. A001620, A297402, A370181, A370182.
Cf. A370565.

Programs

  • Mathematica
    a[n_] := Module[{e2 = IntegerExponent[n, 2], e3 = IntegerExponent[n, 3]}, 2^If[e2 == 0, 0, 1] * 3^e3]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)
  • PARI
    a(n,{p=3}) = my(e = valuation(n, p)); p^e*gcd(p-1, n/p^e)

Formula

Multiplicative with a(3^e) = 3^e, a(2^e) = 2 and a(p^e) = 1 for primes p != 2, 3.
From Amiram Eldar, May 20 2024: (Start)
Dirichlet g.f.: (1 + 1/2^s) * ((1 - 1/3^s)/(1 - 1/3^(s-1))) * zeta(s).
Sum_{k=1..n} a(k) ~ (n/log(3)) * (log(n) + gamma - 1 + log(3) - log(2)/3), where gamma is Euler's constant (A001620). (End)

A370564 Size of the group Q_2*/(Q_2*)^n, where Q_2 is the field of 2-adic numbers.

Original entry on oeis.org

1, 8, 3, 32, 5, 24, 7, 128, 9, 40, 11, 96, 13, 56, 15, 512, 17, 72, 19, 160, 21, 88, 23, 384, 25, 104, 27, 224, 29, 120, 31, 2048, 33, 136, 35, 288, 37, 152, 39, 640, 41, 168, 43, 352, 45, 184, 47, 1536, 49, 200, 51, 416, 53, 216, 55, 896, 57, 232, 59, 480, 61, 248, 63, 8192
Offset: 1

Views

Author

Jianing Song, Apr 30 2024

Keywords

Comments

We have Q_2* = 2^Z X Z_2*, so Q_2*/(Q_2*)^k = (2^Z/2^(kZ)) X (Z_p*/(Z_2*)^k). Note that 2^Z/2^(kZ) is a cyclic group of order k. For the group structure of (Z_2*/(Z_2*)^k), see A370050.

Links

Crossrefs

Row 1 of A370067. Cf. A001620, A370050, A370565, A370566, A370567.
Cf. A297402.

Programs

  • Mathematica
    a[n_] := Module[{e = IntegerExponent[n, 2]}, 2^If[e == 0, 0, e + 1] * n]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)
  • PARI
    a(n) = my(e = valuation(n, 2)); n * 2^e * if(e>=1, 2, 1)

Formula

If n is odd, then a(n) = n. If n = 2^e * n' is even, where n' is odd, then a(n) = n * 2^(e+1).
Multiplicative with a(2^e) = 2^(2*e+1).
a(n) = n * A297402(n).
From Amiram Eldar, May 20 2024: (Start)
Dirichlet g.f.: ((1 - 1/2^(s-1)) * (1 + 1/2^(s-2)) / (1 - 1/2^(s-2))) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (n^2/(2*log(2))) * (log(n) + gamma - 1/2 + log(2)), where gamma is Euler's constant (A001620). (End)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.