A000188 -id:A000188 - cp's OEIS frontend

A055071 Largest square dividing n!.

1, 1, 1, 4, 4, 144, 144, 576, 5184, 518400, 518400, 2073600, 2073600, 101606400, 914457600, 14631321600, 14631321600, 526727577600, 526727577600, 52672757760000, 52672757760000, 6373403688960000, 6373403688960000, 917770131210240000, 22944253280256000000

Offset: 1

Labos Elemer, Jun 13 2000

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Cf. A000188, A008833, A055772.

seq(expand(numtheory[nthpow](n!, 2)), n=1..26); # Peter Luschny, Apr 03 2013

a[n_] := Select[Reverse @ Divisors[n!], IntegerQ[Sqrt[#]] &, 1] // First; a /@ Range[23] (* Jean-François Alcover, May 19 2011 *)
f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n!]); Array[a, 30] (* Amiram Eldar, Jul 26 2024 *)

a(n)=core(n!,2)[2]^2 \\ Charles R Greathouse IV, Apr 04 2012

from math import prod
from itertools import count, islice
from collections import Counter
from sympy import factorint
def A055071_gen(): # generator of terms
    c = Counter()
    for i in count(1):
        c += Counter(factorint(i))
        yield prod(p**(e-(e&1)) for p, e in c.items())
A055071_list = list(islice(A055071_gen(),30)) # Chai Wah Wu, Jul 27 2024

a(n) = A008833(n!).
log a(n) ~ n log n. - Charles R Greathouse IV, Apr 04 2012
a(n) = A055772(n)^2. - Amiram Eldar, Jul 26 2024

More terms from James Sellers, Jun 20 2000

A056056 Square root of largest square dividing n-th central binomial coefficient.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 2, 2, 2, 3, 3, 1, 2, 1, 2, 2, 2, 1, 2, 10, 10, 30, 30, 6, 12, 3, 3, 3, 6, 5, 10, 10, 10, 3, 6, 2, 2, 2, 2, 30, 60, 15, 30, 42, 42, 42, 42, 14, 28, 2, 2, 2, 4, 2, 4, 4, 4, 21, 21, 7, 14, 7, 14, 6, 6, 1, 2, 2, 2, 10, 10, 70, 140, 7, 14, 126, 126, 6, 6, 30, 60

Offset: 1

Labos Elemer, Jul 26 2000

nonn

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

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A056057, A056058, A056059, A056060, A056061.

Table[Sqrt@ Max@ Select[Divisors@ Binomial[n, Floor[n/2]], IntegerQ@ Sqrt@ # &], {n, 0, 86}] (* Michael De Vlieger, Jul 04 2016 *)
a[n_] := Times @@ (First[#]^Floor[Last[#]/2] & /@ FactorInteger[Binomial[n, Floor[n/2]]]); Array[a, 100] (* Amiram Eldar, Sep 06 2020 *)

a(n) = b = binomial(n, n\2); sqrtint(b/core(b)); \\ Michel Marcus, Dec 10 2013

a(n) = A000188(A001405(n)).

A056059 GCD of largest square and squarefree part of central binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 6, 2, 1, 1, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 6, 3, 1, 1, 1, 2, 3, 6, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 3, 6, 6, 3, 1, 2, 2, 1, 2, 1, 3, 6, 1, 1, 1, 2, 1, 2

Offset: 1

Labos Elemer, Jul 26 2000

nonn

			n=14, binomial(14,7) = 3432 = 8*3*11*13. The largest square divisor is 4, squarefree part is 858. So a(14) = gcd(4,858) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000188, A001405, A007913, A008833, A034974, A046098, A055229, A055231.
Cf. A056056, A056057, A056058, A056060, A056061.
A056201 is the cube of this sequence.

Table[GCD[First@ Select[Reverse@ Divisors@ #, IntegerQ@ Sqrt@ # &], Times @@ Power @@@ Map[{#1, Mod[#2, 2]} & @@ # &, FactorInteger@ #]] &@ Binomial[n, Floor[n/2]], {n, 80}] (* Michael De Vlieger, Feb 18 2017, after Zak Seidov at A007913 *)

A001405(n) = binomial(n, n\2);
A055229(n) = { my(c=core(n)); gcd(c, n/c); } \\ Charles R Greathouse IV, Nov 20 2012
A056059(n) = A055229(A001405(n)); \\ Antti Karttunen, Jul 20 2017

from sympy import binomial, gcd
from sympy.ntheory.factor_ import core
def a001405(n): return binomial(n, n//2)
def a055229(n):
    c = core(n)
    return gcd(c, n//c)
def a(n): return a055229(a001405(n))
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Jul 20 2017

a(n) = A055229(A001405(n)), where A055229(n) = gcd(A008833(n), A007913(n)).

Formula clarified by Antti Karttunen, Jul 20 2017

A063775 Number of 4th powers dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Aug 16 2001

			a(79) = 1 since 79 is divisible by 1 = 1^4.
a(80) = 2 since 80 is divisible by 1 and 16 = 2^4.
a(81) = 2 since 81 is divisible by 1 and 81 = 3^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harry J. Smith)

Cf. A046951 (number of squares), A061704 (number of cubes).

Cf. A000005, A053164, A046951, A000188.

seq(coeff(series(add(x^(k^4)/(1-x^(k^4)),k=1..n),x,n+1), x, n), n = 1 .. 120); # Muniru A Asiru, Dec 29 2018

nn = 100;f[list_, i_] := list[[i]];
Table[DirichletConvolve[f[Boole[Map[IntegerQ[#] &, Map[#^(1/4) &,Range[nn]]]], n],f[Table[1, {nn}], n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 07 2015 *)
f[p_, e_] := 1 + Floor[e/4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

{ for (n=1, 2000, k=2; a=1; while ((p=k^4) <= n, if (n%p == 0, a++); k++); write("b063775.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 30 2009

a(n) = A000005(A053164(n)) = A046951(A000188(n)).
Multiplicative with a(p^e) = 1 + floor(e/4).
Dirichlet g.f.: zeta^2(4s)*Product_{primes p} (1 + p^(-s) + p^(-2s) + p^(-3s)). - R. J. Mathar, Jan 11 2012
G.f.: Sum_{k>=1} x^(k^4)/(1 - x^(k^4)). - Ilya Gutkovskiy, Mar 21 2017
Dirichlet g.f.: zeta(s) * zeta(4s). - Álvar Ibeas, Dec 29 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n / 90 + Zeta(1/4) * n^(1/4). - Vaclav Kotesovec, Feb 03 2019

A064179 Infinitary version of Moebius function: infinitary MoebiusMu of n, equal to mu(n) iff mu(n) differs from zero, else 1 or -1 depending on whether the sum of the binary digits of the exponents in the prime decomposition of n is even or odd.

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1

Offset: 1

Wouter Meeussen, Sep 20 2001

Apparently the (ordinary) Dirichlet inverse of A050377. - R. J. Mathar, Jul 15 2010

Also analog of Liouville's function (A008836) in Fermi-Dirac arithmetic, where the terms of A050376 play the role of primes (see examples). - Vladimir Shevelev, Oct 28 2013.

			G.f. = x - x^2 - x^3 - x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 + ...
mu[45]=0 but iMoebiusMu[45]=1 because 45 = 3^2 * 5^1 and the binary digits of 2 and 1 add up to 2, an even number.
A unique representation of 48 over distinct terms of A050376 is 3*16. Since it contains even factors, then a(48)=1; for 54 such a representation is 2*3*9, thus a(54)=-1. - _Vladimir Shevelev_, Oct 28 2013

Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian)

Antti Karttunen, Table of n, a(n) for n = 1..65537
G. L. Cohen, On an integers' infinitary divisors, Math. Comp. 54 (1990), 395-411.
G. L. Cohen and P. Hagis, Jr, Arithmetic functions associated with the infinitary divisors of an integer, Internat. J. Math. Math. Sci. 16 (2) (1993), 373-384.
Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
Rasa Steuding, Jörn Steuding, and László Tóth, A modified Möbius mu-function, Rendiconti del Circolo Matematico di Palermo, Vol. 60 (2011), pp. 13-21; arXiv preprint, arXiv:1109.4242 [math.NT], 2011.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to binary expansion of n.

Sequences with related definitions: A008683, A008836, A064547, A302777.
Positions of -1: A000028.
Positions of 1: A000379.
Cf. A000120, A001221, A050376, A050377, A091732.
Sequences used to express relationships between the terms: A000188, A003961, A007913, A008833, A059897, A225546.

iMoebiusMu[n_] := Switch[MoebiusMu[n], 1, 1, -1, -1, 0, If[OddQ[Plus@@(DigitCount[Last[Transpose[FactorInteger[n]]], 2, 1])], -1, 1]];
(* The Moebius inversion formula seems to hold for iMoebiusMu and the infinitary_divisors of n: if g[ n_ ] := Plus@@(f/@iDivisors[ n ]) for all n, then f[ n_ ]===Plus@@(iMoebiusMu[ # ]g[ n/# ]&/@iDivisors[ n ]) *)
f[p_, e_] := (-1)^DigitCount[e, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 23 2023 *)

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; (-1) ^ subst( Pol( binary(e)), x, 1)))}; /* Michael Somos, Jan 08 2008 */

a(n) = if (n==1, 1, (-1)^omega(core(n)) * a(core(n,1)[2])) \\ Peter Munn, Mar 16 2022

a(n) = vecprod(apply(x -> (-1)^hammingweight(x), factor(n)[, 2])); \\ Amiram Eldar, Dec 23 2023

from math import prod
from sympy import factorint
def A064179(n): return prod(-1 if e.bit_count()&1 else 1 for e in factorint(n).values()) # Chai Wah Wu, Oct 12 2024

(define (A064179 n) (expt -1 (A064547 n))) ;; Antti Karttunen, Nov 23 2017

From Vladimir Shevelev Feb 20 2011: (Start)
Sum_{d runs through i-divisors of n} a(d)=1 if n=1, or 0 if n>1; Sum_{d runs through i-divisors of n} a(d)/d = A091732(n)/n.
Infinitary Moebius inversion:
If Sum_{d runs through i-divisors of n} f(d)=F(n), then f(n) = Sum_{d runs through i-divisors of n} a(d)*F(n/d). (End)
a(n) = (-1)^A064547(n). - R. J. Mathar, Apr 19 2011
Let k=k(n) be the number of terms of A050376 that divide n with odd maximal exponent. Then a(n) = (-1)^k. For example, if n=96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus k(96)=3 and a(96)=-1. - Vladimir Shevelev, Oct 28 2013
From Peter Munn, Jan 25 2020: (Start)
a(A050376(n)) = -1; a(A059897(n,k)) = a(n) * a(k).
a(n^2) = a(n).
a(A003961(n)) = a(n).
a(A225546(n)) = a(n).
a(A000028(n)) = -1; a(A000379(n)) = 1.
(End)
a(n) = a(A007913(n)) * a(A008833(n)) = (-1)^A001221(A007913(n)) * a(A000188(n)). - Peter Munn, Mar 16 2022
From Amiram Eldar, Dec 23 2023: (Start)
Multiplicative with a(p^e) = (-1)^A000120(e).
Dirichlet g.f.: 1/Product_{k>=0} zeta(2^k * s) (Steuding et al., 2011). (End)

A056057 The largest square which divides n-th central binomial coefficient.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 4, 9, 9, 1, 4, 1, 4, 4, 4, 1, 4, 100, 100, 900, 900, 36, 144, 9, 9, 9, 36, 25, 100, 100, 100, 9, 36, 4, 4, 4, 4, 900, 3600, 225, 900, 1764, 1764, 1764, 1764, 196, 784, 4, 4, 4, 16, 4, 16, 16, 16, 441, 441, 49, 196, 49, 196, 36, 36, 1, 4

Offset: 1

Labos Elemer, Jul 26 2000

nonn

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

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A046098, A034974.

Cf. A056056, A056058, A056059, A056060, A056061.

Table[First@ Select[Reverse@ Divisors@ Binomial[n, Floor[n/2]], IntegerQ@ Sqrt@ # &], {n, 72}] (* Michael De Vlieger, Feb 18 2017 *)
a[n_] := Times @@ (First[#]^(2*Floor[Last[#]/2]) & /@ FactorInteger[Binomial[n, Floor[n/2]]]); Array[a, 100] (* Amiram Eldar, Sep 06 2020 *)

a(n) = A008833(A001405(n)).

a(A046098(n)) = 1.

A056626 Number of non-unitary square divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Labos Elemer, Aug 08 2000

nonn

			n = p^u prime power has u+1 square divisors of which 2 (i.e., 1 and n) are unitary but u-1 are not unitary, so a(p^u) = u - 1. E.g., n = 4^4 = 256, has 5 square divisors {1, 4, 16, 64, 256} of which {4, 16, 64} are not unitary, so a(256)=3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000188, A008833, A034444, A046951, A055229, A056624, A162641.

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, ! CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n])- Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Sep 26 2022 *)

a(n) = {my(f = factor(n), r=0, m = 0); prod(i=1,#f~,f[i,2]>>1 + 1) - 2^(omega(f) - omega(core(f)))} \\ David A. Corneth, Jul 28 2017

a(n) = sumdiv(n, d, if(gcd(d, n/d)!=1, issquare(d))); \\ Michel Marcus, Jul 29 2017

from math import prod
from sympy import factorint
def A056626(n):
    f = factorint(n).values()
    return prod((e>>1)+1 for e in f)-(1<Chai Wah Wu, Aug 04 2024

a(n) = A046951(n) - 2^r(n), where r(n) is the number of distinct prime factors of the largest unitary square divisor of n. [Corrected by Amiram Eldar, Aug 03 2024]
a(n) = A046951(n) - 2^(A162641(n)). -  David A. Corneth, Jul 28 2017
From Amiram Eldar, Sep 26 2022: (Start)
a(n) = A046951(n) - A056624(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)*(1 - 1/zeta(3)) = 0.27650128922802056073... . (End)

a(32) and a(96) corrected by Michael De Vlieger, Jul 29 2017

A015052 a(n) is the smallest positive integer m such that m^5 is divisible by n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 4, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset: 1

R. Muller

A multiplicative companion function n/a(n) = 1,1,1,2,1,1,1,4,3,1,1,2,1,1,1,8,1,... can be defined using the 5th instead of the 4th power in A000190, which differs from A000190 and also from A003557. - R. J. Mathar, Jul 14 2012

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
Henry Ibstedt, Surfing on the Ocean of Numbers, Erhus Univ. Press, Vail, 1997.
Florentin Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse, Bucharest, 1996.
Eric Weisstein's World of Mathematics, Smarandache Ceil Function.

Cf. A000188 (inner square root), A019554 (outer square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A053166 (outer 4th root), A015053 (outer 6th root).

Cf. A013667.

f[p_, e_] := p^Ceiling[e/5]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = ceil(f[i,2]/5)); factorback(f); \\ Michel Marcus, Feb 15 2015

Multiplicative with a(p^e) = p^(ceiling(e/5)). - Christian G. Bower, May 16 2005

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(9)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6 + 1/p^7 - 1/p^8) = 0.3523622369... . - Amiram Eldar, Oct 27 2022

Corrected by David W. Wilson, Jun 04 2002

Name reworded by Jon E. Schoenfield, Oct 28 2022

A015053 Smallest positive integer for which n divides a(n)^6.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset: 1

R. Muller (Research37(AT)aol.com)

Differs from A007947 as follows: A007947(128)=2, a(128)=4; A007947(256)=2, a(256)=4; A007947(384)=6, a(384)=12; A007947(512)=2, a(512)=4; A007947(640)=10, a(640)=20, etc. - R. J. Mathar, Oct 28 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
Henry Ibstedt, Surfing on the Ocean of Numbers, Erhus Univ. Press, Vail, 1997.
Eric Weisstein's World of Mathematics, Smarandache Ceil Function.

Cf. A000188 (inner square root), A019554 (outer square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A053166 (outer 4th root), A015052 (5th outer root).

Cf. A013669.

spi[n_]:=Module[{k=1},While[PowerMod[k,6,n]!=0,k++];k]; Array[spi,80] (* Harvey P. Dale, Feb 29 2020 *)
f[p_, e_] := p^Ceiling[e/6]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = ceil(f[i,2]/6)); factorback(f); \\ Michel Marcus, Feb 15 2015

Multiplicative with a(p^e) = p^ceiling(e/6). - Christian G. Bower, May 16 2005

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(11)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6 + 1/p^7 - 1/p^8 + 1/p^9 - 1/p^10) = 0.3522558764... . - Amiram Eldar, Oct 27 2022

Corrected by David W. Wilson, Jun 04 2002

A056061 Number of square divisors of central binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 8, 8, 4, 6, 2, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 2, 2, 2, 8, 12, 4, 8, 8, 8, 8, 8, 4, 6, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 2, 4, 2, 4, 4, 4, 1, 2, 2, 2, 4, 4, 8, 12, 2, 4, 12, 12, 4, 4, 8, 12, 12, 12, 4, 6, 8, 12, 12, 12, 8, 16, 8, 8, 6

Offset: 1

Labos Elemer Jul 26 2000

nonn

			n=27: binomial(27,13) = 20058300, its largest square-divisor is 900=30^2 so a(27) = tau(30) = 8.

Michel Marcus, Table of n, a(n) for n = 1..120

Cf. A001405, A046951, A000005, A000188, A056056, A056057, A056058, A056059, A056060.

Table[Count[Divisors@ Binomial[n, Floor[n/2]], d_ /; IntegerQ@ Sqrt@ d], {n, 0, 84}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = sumdiv(binomial(n, n\2), d, issquare(d)); \\ Michel Marcus, Feb 19 2017

a(n) = A046951(A001405(n)) = A000005(A000188(A001405(n))).

cp's OEIS Frontend

A055071 Largest square dividing n!.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A056056 Square root of largest square dividing n-th central binomial coefficient.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A056059 GCD of largest square and squarefree part of central binomial coefficients.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A063775 Number of 4th powers dividing n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A064179 Infinitary version of Moebius function: infinitary MoebiusMu of n, equal to mu(n) iff mu(n) differs from zero, else 1 or -1 depending on whether the sum of the binary digits of the exponents in the prime decomposition of n is even or odd.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Python

Scheme

Formula

A056057 The largest square which divides n-th central binomial coefficient.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A056626 Number of non-unitary square divisors of n.

Views

Author

Keywords

Examples