A370180 -id:A370180 - cp's OEIS frontend

A370050 Square array read by ascending antidiagonals: T(n,k) is the size of the group Z_p/(Z_p)^k, where p = prime(n), and Z_p is the ring of p-adic integers.

1, 1, 4, 1, 2, 1, 1, 2, 3, 8, 1, 2, 1, 2, 1, 1, 2, 3, 4, 1, 4, 1, 2, 1, 2, 5, 6, 1, 1, 2, 3, 2, 1, 2, 1, 16, 1, 2, 1, 4, 5, 6, 1, 2, 1, 1, 2, 3, 4, 1, 2, 7, 4, 9, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 3, 10, 1, 8, 1, 2, 3, 4, 1, 6, 1, 4, 1, 2, 1, 6, 1

Offset: 1

Jianing Song, Apr 30 2024

We have that Z_p*/(Z_p*)^k is the inverse limit of (Z/p^iZ)*/((Z/p^iZ)*)^k as i tends to infinity. Write k = p^e * k' with k' not being divisible by p. If p is odd, then the group is cyclic of order p^e * gcd(p-1,k'). If p = 2 and k is odd, then the group is trivial. If p = 2 and k is even, then the group is the product of a cyclic group of order 2^e and a cyclic group of order 2.

Each row is multiplicative.

			Table reads
  1, 4, 1, 8, 1, 4, 1, 16, 1, 4
  1, 2, 3, 2, 1, 6, 1, 2, 9, 2
  1, 2, 1, 4, 5, 2, 1, 4, 1, 10
  1, 2, 3, 2, 1, 6, 7, 2, 3, 2
  1, 2, 1, 2, 5, 2, 1, 2, 1, 10
  1, 2, 3, 4, 1, 6, 1, 4, 3, 2
  1, 2, 1, 4, 1, 2, 1, 8, 1, 2
  1, 2, 3, 2, 1, 6, 1, 2, 9, 2
  1, 2, 1, 2, 1, 2, 1, 2, 1, 2
  1, 2, 1, 4, 1, 2, 7, 4, 1, 2
For p = prime(1) = 2 and k = 2, we have Z_p*/(Z_p*)^k = Z_2*/(1+8Z_2) = (Z/8Z)*/(1+8Z) = C_2 X C_2, so T(1,2) = 4.
For p = prime(2) = 3 and k = 3, we have Z_p*/(Z_p*)^k = Z_3*/((1+9Z_3) U (8+9Z_3)) = (Z/9Z)*/((1+9Z) U (8+9Z)) = C_3, so T(2,3) = 3.

Jianing Song, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals)

Cf. A370067. Row 1-4: A297402, A370180, A370181, A370182.

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

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

A370565 Size of the group Q_3/(Q_3)^n, where Q_3 is the field of 3-adic numbers.

Original entry on oeis.org

1, 4, 9, 8, 5, 36, 7, 16, 81, 20, 11, 72, 13, 28, 45, 32, 17, 324, 19, 40, 63, 44, 23, 144, 25, 52, 729, 56, 29, 180, 31, 64, 99, 68, 35, 648, 37, 76, 117, 80, 41, 252, 43, 88, 405, 92, 47, 288, 49, 100, 153, 104, 53, 2916, 55, 112, 171, 116, 59, 360, 61, 124, 567, 128

Offset: 1

Jianing Song, Apr 30 2024

We have Q_3* = 3^Z X Z_3*, so Q_3*/(Q_3*)^k = (3^Z/3^(kZ)) X (Z_p*/(Z_3*)^k). Note that 3^Z/3^(kZ) is a cyclic group of order k. For the group structure of (Z_3*/(Z_3*)^k), see A370050.

Jianing Song, Table of n, a(n) for n = 1..10000

Row 2 of A370067. Cf. A001620, A370050, A370564, A370566, A370567.

Cf. A370180.

a[n_] := Module[{e2 = IntegerExponent[n, 2], e3 = IntegerExponent[n, 3]}, 2^Min[e2, 1] * 3^e3 * n]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

a(n, {p=3}) = my(e = valuation(n, p)); n * p^e*gcd(p-1, n/p^e)

Write n = 3^e * n' with k' not being divisible by 3, then a(n) = n * 3^e * gcd(2,n').
Multiplicative with a(3^e) = 3^(2*e), a(2^e) = 2^(e+1) and a(p^e) = p^e for primes p != 2, 3.
a(n) = n * A370180(n).
From Amiram Eldar, May 20 2024: (Start)
Dirichlet g.f.: ((1 + 1/2^(s-1)) * (1 - 1/3^(s-1))/(1 - 1/3^(s-2))) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (n^2/(2*log(3))) * (log(n) + gamma - 1/2 + log(3) - log(2)/3), where gamma is Euler's constant (A001620). (End)

A370181 Size of the group Z_5/(Z_5)^n, where Z_5 is the ring of 5-adic integers.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 4, 1, 10, 1, 4, 1, 2, 5, 4, 1, 2, 1, 20, 1, 2, 1, 4, 25, 2, 1, 4, 1, 10, 1, 4, 1, 2, 5, 4, 1, 2, 1, 20, 1, 2, 1, 4, 5, 2, 1, 4, 1, 50, 1, 4, 1, 2, 5, 4, 1, 2, 1, 20, 1, 2, 1, 4, 5, 2, 1, 4, 1, 10, 1, 4, 1, 2, 25, 4, 1, 2, 1, 20, 1, 2, 1, 4, 5, 2, 1, 4, 1, 10

Offset: 1

Jianing Song, Apr 30 2024

We have that Z_5*/(Z_5*)^n is the inverse limit of (Z/5^iZ)*/((Z/5^iZ)*)^n as i tends to infinity. Write n = 5^e * n' with n' not being divisible by 5, then the group is cyclic of order 5^e * gcd(4,n'). See A370050.

			We have Z_5*/(Z_5*)^5 = Z_5* / ((1+25Z_5) U (7+25Z_5) U (18+25Z_5) U (24+25Z_5)) = (Z/25Z)*/((1+25Z) U (7+25Z) U (18+25Z) U (24+25Z)) = C_5, so a(5) = 5.
We have Z_5*/(Z_5*)^10 = Z_5* / ((1+25Z_5) U (24+25Z_5)) = (Z/25Z)*/((1+25Z) U (25+25Z)) = C_10, so a(10) = 10.

Jianing Song, Table of n, a(n) for n = 1..10000

Row 3 of A370050. Cf. A001620, A297402, A370180, A370182.

Cf. A370566.

a[n_] := Module[{e2 = IntegerExponent[n, 2], e5 = IntegerExponent[n, 5]}, 2^Min[e2, 2] * 5^e5]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

a(n,{p=5}) = my(e = valuation(n, p)); p^e*gcd(p-1, n/p^e)

Multiplicative with a(5^e) = 5^e, a(2) = 2, a(2^e) = 4 for e >= 2 and a(p^e) = 1 for primes p != 2, 5.
From Amiram Eldar, May 20 2024: (Start)
Dirichlet g.f.: (1 + 1/2^s + 1/2^(2*s-1)) * ((1 - 1/5^s)/(1 - 1/5^(s-1))) * zeta(s).
Sum_{k=1..n} a(k) ~ (8*n/(5*log(5))) * (log(n) + gamma - 1 + (3/4)*log(5/2)), where gamma is Euler's constant (A001620). (End)

A370182 Size of the group Z_7/(Z_7)^n, where Z_7 is the ring of 7-adic integers.

Original entry on oeis.org

1, 2, 3, 2, 1, 6, 7, 2, 3, 2, 1, 6, 1, 14, 3, 2, 1, 6, 1, 2, 21, 2, 1, 6, 1, 2, 3, 14, 1, 6, 1, 2, 3, 2, 7, 6, 1, 2, 3, 2, 1, 42, 1, 2, 3, 2, 1, 6, 49, 2, 3, 2, 1, 6, 1, 14, 3, 2, 1, 6, 1, 2, 21, 2, 1, 6, 1, 2, 3, 14, 1, 6, 1, 2, 3, 2, 7, 6, 1, 2, 3, 2, 1, 42, 1, 2, 3, 2, 1, 6

Offset: 1

Jianing Song, Apr 30 2024

We have that Z_7*/(Z_7*)^n is the inverse limit of (Z/7^iZ)*/((Z/7^iZ)*)^n as i tends to infinity. Write n = 7^e * n' with n' not being divisible by 7, then the group is cyclic of order 7^e * gcd(7,n'). See A370050.

			We have Z_7*/(Z_7*)^7 = Z_7* / ((1+49Z_7) U (18+49Z_7) U (19+49Z_7) U (30+49Z_7) U (31+49Z_7) U (48+49Z_7)) = (Z/49Z)*/((1+49Z) U (18+49Z) U (19+49Z) U (30+49Z) U (31+49Z) U (48+49Z)) = C_7, so a(7) = 7.
We have Z_7*/(Z_7*)^14 = Z_7* / ((1+49Z_7) U (18+49Z_7) U (30+49Z_7)) = (Z/49Z)*/((1+49Z) U (18+49Z) U (30+49Z)) = C_14, so a(14) = 14.

Jianing Song, Table of n, a(n) for n = 1..10000

Row 4 of A370050. Cf. A001620, A297402, A370180, A370181.

Cf. A370567.

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]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

a(n,{p=7}) = my(e = valuation(n, p)); p^e*gcd(p-1, n/p^e)

Multiplicative with a(7^e) = 7^e, a(2^e) = 2, a(3^e) = 3 and a(p^e) = 1 for primes p != 2, 3, 7.
From Amiram Eldar, May 20 2024: (Start)
Dirichlet g.f.: (1 + 1/2^s) * (1 + 2/3^s) * ((1 - 1/7^s)/(1 - 1/7^(s-1))) * zeta(s).
Sum_{k=1..n} a(k) ~ (15*n/(7*log(7))) * (log(n) + gamma - 1 + 2*log(7)/3 - 2*log(3)/5 - log(2)/3), where gamma is Euler's constant (A001620). (End)

cp's OEIS Frontend

A370050 Square array read by ascending antidiagonals: T(n,k) is the size of the group Z_p*/(Z_p*)^k, where p = prime(n), and Z_p is the ring of p-adic integers.

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A370565 Size of the group Q_3*/(Q_3*)^n, where Q_3 is the field of 3-adic numbers.

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A370181 Size of the group Z_5*/(Z_5*)^n, where Z_5 is the ring of 5-adic integers.

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A370182 Size of the group Z_7*/(Z_7*)^n, where Z_7 is the ring of 7-adic integers.

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A370050 Square array read by ascending antidiagonals: T(n,k) is the size of the group Z_p/(Z_p)^k, where p = prime(n), and Z_p is the ring of p-adic integers.

A370565 Size of the group Q_3/(Q_3)^n, where Q_3 is the field of 3-adic numbers.

A370181 Size of the group Z_5/(Z_5)^n, where Z_5 is the ring of 5-adic integers.

A370182 Size of the group Z_7/(Z_7)^n, where Z_7 is the ring of 7-adic integers.