Matthias Kaak - cp's OEIS frontend

User: Matthias Kaak

Matthias Kaak has authored 2 sequences.

A356473 Denominator of the average of gcd(i,n) for i = 1..n.

1, 2, 3, 1, 5, 2, 7, 2, 3, 10, 11, 3, 13, 14, 1, 1, 17, 2, 19, 5, 21, 22, 23, 6, 5, 26, 1, 7, 29, 2, 31, 2, 11, 34, 35, 3, 37, 38, 39, 2, 41, 14, 43, 11, 5, 46, 47, 1, 7, 10, 17, 13, 53, 2, 55, 14, 57, 58, 59, 1, 61, 62, 3, 1, 13, 22, 67, 17, 23, 70, 71, 6, 73, 74, 3, 19, 11, 26, 79, 5, 3, 82, 83, 21, 85, 86, 29

Offset: 1

Matthias Kaak, Aug 08 2022

From Robert Israel, Dec 29 2022: (Start)
If n is prime, a(n) = n.
If n is an odd prime, a(n^2) = n.
If p and q are distinct primes with p | 2*q-1, then a(p*q) = q, except in the case of 3*5, where both 3 | 2*5-1 and 5 | 2*3-1, and a(3*5) = 1.
For semiprimes p*q where p <> q, p does not divide 2*q-1 and q does not divide 2*p-1, a(p*q) = p*q. (End)

			For n = 3, the average of the gcd's is (gcd(1,3) + gcd(2,3) + gcd(3,3))/3 = (1 + 1 + 3)/3 = 5/3 which has denominator a(3)=3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A356472 (numerators), A018804.

map denominator (map (\i -> sum (map (\j -> gcd i j) [1..i]) % i) [1..])

f:= proc(n) local i;  denom(add(igcd(i,n),i=1..n)/n) end proc:
map(f, [$1..100]); # Robert Israel, Dec 29 2022

Table[Denominator[Sum[GCD[I, j], {j, 1, I}]/I], {I, 100}]
f[p_, e_] := e*(p - 1)/p + 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = denominator(sum(i=1, n, gcd(i, n))/n); \\ Michel Marcus, Aug 08 2022

a(n,f=factor(n))=n/gcd(prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2]),n) \\ Charles R Greathouse IV, Sep 08 2022

from math import gcd, prod
from sympy import factorint
def A356473(n): return (p:=prod(f:=factorint(n)))//gcd(p,prod((p-1)*e+p for p, e in f.items())) # Chai Wah Wu, Sep 08 2022

a(n) = denominator of A018804(n)/n.

a(n) divides n, so in particular a(n) <= n. - Charles R Greathouse IV, Sep 08 2022

A356472 Numerator of the average of gcd(i,n) for i = 1..n.

Original entry on oeis.org

1, 3, 5, 2, 9, 5, 13, 5, 7, 27, 21, 10, 25, 39, 3, 3, 33, 7, 37, 18, 65, 63, 45, 25, 13, 75, 3, 26, 57, 9, 61, 7, 35, 99, 117, 14, 73, 111, 125, 9, 81, 65, 85, 42, 21, 135, 93, 5, 19, 39, 55, 50, 105, 9, 189, 65, 185, 171, 117, 6, 121, 183, 13, 4, 45, 105, 133, 66, 75, 351, 141, 35, 145, 219, 13, 74, 39, 125, 157

Offset: 1

Matthias Kaak, Aug 08 2022

			For n = 3, the average of the gcd's is (gcd(1,3) + gcd(2,3) + gcd(3,3))/3 = (1 + 1 + 3)/3 = 5/3 and its numerator is a(3)=5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A356473 (denominators), A018804, A057661 (LCM).

Cf. A001620, A013661, A306016.

map numerator (map (\i -> sum (map (\j -> gcd i j) [1..i]) % i) [1..])

Table[Numerator[Sum[GCD[I, j], {j, 1, I}]/I], {I, 100}]
f[p_, e_] := e*(p - 1)/p + 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = numerator(sum(i=1, n, gcd(i, n))/n); \\ Michel Marcus, Aug 08 2022

a(n,f=factor(n))=my(k=prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2])); k/gcd(k,n) \\ Charles R Greathouse IV, Sep 08 2022

from math import prod, gcd
from sympy import factorint
def A356472(n):
    f = factorint(n)
    return (m:=prod((p-1)*e+p for p, e in f.items()))//gcd(prod(f),m) # Chai Wah Wu, Sep 08 2022

a(n) = numerator(A018804(n)/n).
a(n) << n^(1+e) for any e > 0. a(n) > 1 for all n > 1. - Charles R Greathouse IV, Sep 08 2022
Sum_{k=1..n} a(k)/A356473(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2024

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User: Matthias Kaak

A356473 Denominator of the average of gcd(i,n) for i = 1..n.

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A356472 Numerator of the average of gcd(i,n) for i = 1..n.

Views

Author

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Examples

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Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Formula