A056622 -id:A056622 - cp's OEIS frontend

A056627 a(n) = A056622(n!).

1, 1, 1, 1, 1, 12, 12, 12, 36, 720, 720, 480, 480, 1680, 3024, 12096, 12096, 145152, 145152, 7257600, 345600, 1900800, 1900800, 136857600, 684288000, 4447872000, 4447872000, 435891456000, 435891456000, 3138418483200, 3138418483200, 6276836966400, 190207180800

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name "Square root of largest unitary square divisor of n!" was incorrect. See A374989 for the correct sequence with this name. - Amiram Eldar, Jul 26 2024

			a(12) = A056622(12!) = A000188(12!)/A055229(12!) = 1440/3 = 480.

Cf. A055772, A055230, A000188, A055229, A056622, A056628, A374989.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 33] (* Amiram Eldar, Jul 08 2024 *)

A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A008833(n) = n/core(n) \\ Michael B. Porter, Oct 17 2009
A056623(n) = (A008833(n)/(A055229(n)^2)); \\ Antti Karttunen, Nov 19 2017
a(n) = sqrtint(A056623(n!)); \\ Michel Marcus, Aug 16 2020

a(n) = A055772(n)/A055230(n) = A000188(n!)/A055229(n!).
a(n) = A056622(n!). - Michel Marcus, Aug 16 2020
a(n) = sqrt(A056628(n)). - Amiram Eldar, Jul 08 2024

More terms from Michel Marcus, Aug 16 2020

Incorrect name replaced with a formula by Amiram Eldar, Jul 26 2024

A056646 a(n) = A056622(A001405(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 2, 2, 1, 3, 3, 1, 2, 1, 2, 2, 1, 1, 2, 10, 5, 10, 5, 3, 12, 3, 3, 1, 2, 5, 10, 10, 5, 3, 6, 2, 1, 2, 1, 5, 20, 15, 30, 42, 21, 14, 7, 7, 28, 2, 1, 1, 4, 1, 4, 4, 2, 21, 21, 7, 14, 7, 14, 6, 3, 1, 2, 2, 1, 10, 5, 35, 140, 7, 14, 126, 63, 2, 1, 5, 20, 90, 45, 3, 12

Offset: 1

Labos Elemer, Aug 09 2000

nonn

Previous name "Square root of largest unitary square divisor of central binomial coefficient" was incorrect. See A376554 for the correct sequence with this name. - Amiram Eldar, Sep 28 2024

			a(28) = A056622(binomial(28,14)) = A056622(40116600) = 5.

Cf. A000188, A001405, A055229, A056056, A056059, A056622, A376554.

a(n) = A000188(A001405(n))/A055229(A001405(n)) = A056056(n)/A056059(n).

Incorrect name replaced with a formula by Amiram Eldar, Sep 28 2024

A071974 Numerator of rational number i/j such that Sagher map sends i/j to n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 19 2002

The Sagher map sends Product p_i^e_i / Product q_i^f_i (p_i and q_i being distinct primes) to Product p_i^(2e_i) * Product q_i^(2f_i-1). This is multiplicative.

			The Sagher map sends the following fractions to 1, 2, 3, 4, ...: 1/1, 1/2, 1/3, 2/1, 1/5, 1/6, 1/7, 1/4, 3/1, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David M. Bradley, Counting the Positive Rationals: A Brief Survey, arXiv:math/0509025 [math.HO], 2005.
Gerald Freilich, A denumerability formula for the rationals, Amer. Math. Monthly, Nov 1965, pp. 1013-1014.
Kevin McCrimmon, Enumeration of the positive rationals, Amer. Math. Monthly, Nov 1960, p. 868.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
Index entries for sequences related to enumerating the rationals

Cf. A071975. Differs from A056622 at a(32).
Cf. A000290, A027748, A077591, A124010, A350388.
For other bijective mappings from integers to positive rationals see A002487, A020652/A020653, A038568/A038569, A229994/A077610, A295515.
Cf. A307868.

a071974 n = product $ zipWith (^) (a027748_row n) $
   map (\e -> (1 - e `mod` 2) * e `div` 2) $ a124010_row n
-- Reinhard Zumkeller, Jun 15 2012

f[{p_, a_}] := If[EvenQ[a], p^(a/2), 1]; a[n_] := Times@@(f/@FactorInteger[n])
Table[Sqrt@ SelectFirst[Reverse@ Divisors@ n, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 104}] (* Michael De Vlieger, Dec 06 2018 *)

a(n)=local(v=factor(n)~); prod(k=1,length(v),if(v[2,k]%2,1,v[1,k]^(v[2,k]/2)))

from math import prod
from sympy import factorint
def A071974(n): return prod(p**(e>>1) for p, e in factorint(n).items() if e&1^1) # Chai Wah Wu, Jul 27 2024

If n=Product p_i^e_i, then a_n=Product p_i^f(e_i), where f(n)=n/2 if n is even and f(n)=0 if n is odd. - Reiner Martin, Jul 08 2002
a(n^2) = n, A071975(n^2) = 1, cf. A000290; a(2*(2*n-1)^2) = 2*n+1, A071975(2*(2*n-1)^2) = 2, cf. A077591. - Reinhard Zumkeller, Jul 10 2011
From Amiram Eldar, Nov 02 2023, Jul 26 2024: (Start)
a(n) = sqrt(A350388(n)) (square root of largest unitary divisor of n that is a square).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-1)). (End)
From Vaclav Kotesovec, May 05 2025: (Start)
Let f(s) = Product_{p prime} (1 -  (p^s + p)/((p^s + 1)*p^(2*s))).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A307868 = Product_{p prime} (1 - 2/(p*(1+p))) = 0.4716806136129978680752356330804820874259263820069868836357372554177321...
f'(1) = f(1) * Sum_{p prime} (5*p+3)*log(p) / ((p+1)*(p^2+p-2)) = f(1) * 2.1244279471327068377850377690765768532203174482128717024402373817115555...
and gamma is the Euler-Mascheroni constant A001620. (End)

More terms from Reiner Martin, Jul 08 2002

Additional references supplied by Kevin Ryde added by N. J. A. Sloane, May 31 2012

A056623 If n=LLgggf (see A056192) and a(n) = LL, then its complementary divisor n/LL = gggf and gcd(L^2, n/LL) = 1.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 1, 25, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 1, 49, 9

Offset: 1

Labos Elemer, Aug 08 2000

The part of the name "Largest unitary square divisor of n" was removed because it is correct only for numbers whose odd exponents in their prime factorization are all smaller than 5. For the correct largest unitary square divisor of n see A350388. - Amiram Eldar, Jul 26 2024

			a(200) = A008833(200)/A055229(200)^2 = 100/2^2 = 25.
a(250) = A008833(250)/A055229(250)^2 = 25/5^2 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A008833, A055229, A000188, A046951, A034444, A056192, A056622, A350388.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^e, p^(e-3)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A008833(n) = n/core(n) \\ Michael B. Porter, Oct 17 2009
A056623(n) = (A008833(n)/(A055229(n)^2)); \\ Antti Karttunen, Nov 19 2017

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e == 1, 1, if(e%2, p^(e-3), p^e)));} \\ Amiram Eldar, Dec 25 2023

(definec (A056623 n) (if (= 1 n) n (let ((e (A067029 n)) (rest (A056623 (A028234 n)))) (cond ((even? e) (* (A028233 n) rest)) ((= 1 e) rest) (else (* (expt (A020639 n) (- e 3)) rest)))))) ;; After Jovovic's multiplicative formula, using memoization-macro definec - Antti Karttunen, Nov 19 2017

a(n) = A008833(n)/A055229(n)^2 = K^2/g^2, which coincides with the largest square divisor iff the g-factor is 1.
Multiplicative with a(p^e)=p^e for even e, a(p)=1, a(p^e)=p^(e-3) for odd e > 1. - Vladeta Jovovic, Apr 30 2002
From Amiram Eldar, Dec 25 2023 (Start)
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2) + 1/p^(9/2)) = 1.81133051934001073532... . (End)
a(n) = A056622(n)^2. - Amiram Eldar, Jul 26 2024

Name edited by Amiram Eldar, Jul 26 2024

A375568 a(n) = denominator(A006571(n)/A366450(n)) if A366450(n) != 0, otherwise 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 9, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 14, 5, 1, 2, 1, 9, 1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 4, 27, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 14, 3, 5, 1, 1, 1, 1

Offset: 1

Mats Granvik, Aug 19 2024

a(n) differs from A071974 at n = 27, 32, 36, 49, 54, 72, 76, 81, 96, 98, 100, 108, 116, 125, 135, 144,...
a(n) differs from A056622 at n = 27, 32, 36, 49, 54, 72, 76, 81, 96, 98, 100, 108, 116, 125, 128, 135, 144,...
GCD(a(n), A071974(n)) differs from A071974 at n = 36, 72, 76, 100, 116, 144,...
GCD(a(n), A056622(n)) differs from A056622 at n = 36, 72, 76, 100, 116, 128, 144,...

Cf. A006571, A366450, A056622, A071974.

nn = 104; a[n_] := DivisorSum[n, MoebiusMu[#]   # &]; f = (x^3 - x^2 - y^2 - y); w[n_] := SeriesCoefficient[q*(Product[(1 - q^k), {k, 11, n, 11}]*Product[1 - q^k, {k, n}])^2, {q, 0, n}]; A006571 = ParallelTable[w[n], {n, 1, nn}]; A366450 = ParallelTable[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0]*a[k]/n, {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]; Denominator[A006571/A366450]

cp's OEIS Frontend

A056627 a(n) = A056622(n!).

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A056646 a(n) = A056622(A001405(n)).

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A071974 Numerator of rational number i/j such that Sagher map sends i/j to n.

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A056623 If n=LLgggf (see A056192) and a(n) = LL, then its complementary divisor n/LL = gggf and gcd(L^2, n/LL) = 1.

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A375568 a(n) = denominator(A006571(n)/A366450(n)) if A366450(n) != 0, otherwise 1.

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