A048633 -id:A048633 - cp's OEIS frontend

A331586 Even numbers n such that A048633(n+1) = A048633(n).

174, 398, 474, 934, 1214, 1934, 2254, 2638, 2966, 3806, 3886, 4024, 4574, 4696, 4718, 4928, 4958, 4990, 5014, 5246, 5290, 5438, 6698, 6934, 7028, 7136, 7258, 7266, 7424, 7694, 7838, 8176, 8448, 8574, 8720, 8958, 9854, 9974, 10174, 10334, 10448, 11338, 11374, 12094, 12102, 12220, 12462, 12626

Offset: 1

Robert Israel, Jan 21 2020

nonn

binomial(2k+1,k)/binomial(2k,k) = (2k+1)/(k+1), so 2k is a member if and only if every prime dividing 2k+1 divides binomial(2k,k) and every prime dividing k+1 divides binomial(2k+1,k).

A048633(n+1)=A048633(n) for all odd numbers n except the Mersenne numbers (A000225).

			a(1)=174 is a member because 174 is even and A048633(174)=A048633(175)=632127493640977953733428652337034082437215015190.

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

Cf. A000225, A048633.

g:= proc(m,n,p)
  local Lm, Ln;
   Lm:= convert(m,base,p);
   Ln:= convert(n,base,p);
   min(Lm[1..nops(Ln)]-Ln) < 0
end proc:
filter:= proc(n) local p;
  for p in numtheory:-factorset(n+1) do
     if not g(n,n/2,p) then return false fi;
  od;
  for p in numtheory:-factorset(n/2+1) do
     if not g(n+1,n/2,p) then return false fi
  od;
  true
end proc:
select(filter, [seq(i,i=2..15000,2)]);

A056058 Squarefree part of the n-th central binomial coefficient.

Original entry on oeis.org

1, 2, 3, 6, 10, 5, 35, 70, 14, 7, 462, 231, 429, 858, 715, 1430, 24310, 12155, 92378, 46189, 88179, 176358, 1352078, 676039, 52003, 104006, 22287, 44574, 2154410, 1077205, 33393355, 66786710, 129644790, 64822395, 181502706, 90751353

Offset: 1

Labos Elemer, Jul 26 2000

nonn

a(3387) has 1001 decimal digits. - Michael De Vlieger, Jul 14 2017

Michael De Vlieger, Table of n, a(n) for n = 1..3386

Cf. A048633 for another version.

Table[Sqrt[Binomial[n, Floor[n/2]]] /. (c_: 1) a_^(b_: 0) :> (c a^b)^2, {n, 36}] (* Michael De Vlieger, Jul 14 2017, after Bill Gosper at A007913 *)

a(n) = A007913(A001405(n)).

For squarefree central binomial coefficients (A046098), a(n)=A001405(n).

A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 30, 105, 70, 42, 210, 2310, 2310, 4290, 6006, 15015, 30030, 170170, 510510, 1939938, 1385670, 881790, 9699690, 223092870, 44618574, 17160990, 74364290, 31870410, 223092870, 6469693230, 6469693230, 100280245065

Offset: 0

Labos Elemer, Aug 07 2000

nonn

Also squarefree kernel of A001142; row products in table A256113. - Reinhard Zumkeller, Mar 21 2015
a(2372) has 1001 decimal digits. - Michael De Vlieger, Jul 14 2017
Also the squarefree kernel of the cumulative product of n^n/n!. - Peter Luschny, Dec 21 2019
Conjecture: the few odd values belong to A070826. - Bill McEachen, Jun 24 2023
And their indices appear to be A007053. - Michel Marcus, Jul 01 2023

			a(7) = 105 because lcm(1, 7, 21, 35) = 105 is already squarefree.
a(0) = 1 because n^n/n! = 1 for the integer n = 0. - _Peter Luschny_, Dec 21 2019

Michael De Vlieger, Table of n, a(n) for n = 0..2371

Cf. A002944, A034386, A048633, A003418, A000142, A001405.
Cf. A007947, A001142, A256113, A002109, A000178, A329947.
Cf. A070826.

a056606 = a007947 . a001142  -- Reinhard Zumkeller, Mar 21 2015

h := n -> mul(k^k/factorial(k), k=0..n):
rad := n -> mul(k, k = numtheory[factorset](n)):
seq(rad(h(n)), n=0..31); # Peter Luschny, Dec 21 2019

Table[Apply[Times, FactorInteger[Product[k^(2 k - 1 - n), {k, n}]][[All, 1]]], {n, 0, 31}] (* or *)
Table[Apply[Times, FactorInteger[Apply[LCM, Range@ n]/n][[All, 1]]], {n, 1, 32}] (* Michael De Vlieger, Jul 14 2017 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = rad(lcm(vector(n+1, k, binomial(n,k-1)))); \\ Michel Marcus, Jun 24 2023

a(n) = A007947(A002944(n+1)). - Michel Marcus, Dec 21 2019

a(n) = radical(hyperfactorial(n)/superfactorial(n)) = A007947(A002109(n)/ A000178(n)) for n >= 0. - Peter Luschny, Dec 21 2019

Extended with a(0) = 1 by Peter Luschny, Dec 21 2019

A080397 Largest squarefree number dividing central binomial coefficient A000984(n).

Original entry on oeis.org

1, 2, 6, 10, 70, 42, 462, 858, 4290, 24310, 92378, 176358, 1352078, 520030, 222870, 6463230, 200360130, 129644790, 907513530, 1767263190, 22974421470, 134564468610, 526024740930, 22870640910, 1074920122770, 1504888171878, 1967930686302, 34766775458002, 1912172650190110

Offset: 0

Labos Elemer, Mar 19 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A007947, A000984, A001405, A048633.

a := n -> convert(numtheory:-factorset(binomial(2*n, n)), `*`):
seq(a(n),n=0..25); # Peter Luschny, Oct 31 2015

a[n_] := Times @@ FactorInteger[Binomial[2n, n]][[All, 1]]; Array[a, 26, 0] (* Jean-François Alcover, Jun 04 2019 *)

a(n) = vecprod(factor(binomial(2*n, n))[, 1]); \\ Amiram Eldar, Jun 21 2024

a(n) = A007947(A000984(n)).

More terms from Amiram Eldar, Jun 21 2024

A056607 a(n) is the n-th primorial divided by squarefree kernel of corresponding central binomial coefficient.

Original entry on oeis.org

2, 3, 10, 35, 231, 3003, 14586, 138567, 5311735, 154040315, 434113615, 16062203755, 354604036745, 15247973580035, 286661903304658, 7596540437573437, 79093391614735197, 4824696888498847017, 85067024086690197405, 6039758710155004015755, 230948868774022296411965

Offset: 1

Labos Elemer, Aug 07 2000

nonn

			For n = 6, A002110(6) = 30030, C(6,3) = 20 with squarefree kernel 10. So, a(6) = 30030/10 = 3003 = 3*7*11*13, which is also squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..383

Cf. A002110, A001405, A048633.

a[n_] := Times @@ Prime[Range[n]] / Times @@ FactorInteger[Binomial[n, Floor[n/2]]][[;;, 1]]; Array[a, 21] (* Amiram Eldar, Mar 01 2025 *)

a(n) = prod(k=1, n, prime(k))/factorback(factor(binomial(n, n\2))[, 1]); \\ Michel Marcus, Jul 13 2018

a(n) = A002110(n)/A048633(n).

A056610 Quotient: squarefree kernel of lcm(1,..,n) (or of n!) divided by kernel of central binomial coefficient.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 6, 3, 5, 5, 5, 5, 35, 35, 14, 7, 21, 21, 105, 105, 55, 55, 165, 165, 429, 429, 1001, 1001, 1001, 1001, 2002, 1001, 1547, 1547, 221, 221, 4199, 4199, 323, 323, 2261, 2261, 24871, 24871, 572033, 572033, 572033, 572033, 408595, 408595

Offset: 1

Labos Elemer, Aug 07 2000

nonn

			n = 10, lcm(1, 2, 3, ..., 10) = 2520, lcm(1, 10, 45, 120, 210, 252) = 252; the corresponding kernels are 210 or 42. The quotient is a(10) = 5.

Cf. A003418, A000142, A034386, A007947, A001405, A048633.

a(n) = A007947(A000142(n))/A007947(A001405(n)) = A034386(n)/A048633(n).

A048681 Maximum over k of the largest squarefree number dividing a value of binomial(n,k).

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 35, 70, 42, 210, 462, 462, 858, 3003, 5005, 4290, 24310, 24310, 92378, 125970, 293930, 646646, 1352078, 1352078, 817190, 5311735, 2897310, 13123110, 34597290, 17298645, 100180065, 200360130, 129644790, 2203961430

Offset: 1

Labos Elemer

nonn

			For n=10, the squarefree kernels of binomial(n,k) are {1, 10, 15, 30, 210, 42, 210, 30, 15, 10, 1}, so the maximal largest squarefree divisor is that of binomial(10,4)=210: it is 210, so a(10)=210. (It is not equal to the largest squarefree number dividing binomial(10,5)=252, which is A048633(10)=42.) [edited by _Jon E. Schoenfield_, May 19 2018]

Analogous sequences for A001221, A001222, A000005 are given in A048273, A048275, A048620.

a(n) = vecmax(vector(ceil(n\2)+1, k, factorback(factorint(binomial(n,k-1))[, 1]))); \\ Michel Marcus, May 20 2018

cp's OEIS Frontend

A331586 Even numbers n such that A048633(n+1) = A048633(n).

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A056058 Squarefree part of the n-th central binomial coefficient.

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A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

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A080397 Largest squarefree number dividing central binomial coefficient A000984(n).

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A056607 a(n) is the n-th primorial divided by squarefree kernel of corresponding central binomial coefficient.

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A056610 Quotient: squarefree kernel of lcm(1,..,n) (or of n!) divided by kernel of central binomial coefficient.

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A048681 Maximum over k of the largest squarefree number dividing a value of binomial(n,k).

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