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Search: a056173 -id:a056173

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A056175

Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).

+10
6

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 3, 3, 2, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 0, 1, 1, 1, 2, 2, 3, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,10

COMMENTS

Number of prime divisors of the largest square dividing A001405(n). (A prime divisor is nonunitary iff its exponent exceeds 1.)

LINKS

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

FORMULA

a(n) = A001221(A000188(A001405(n))).

a(n) = A001221(A056057(n)).

EXAMPLE

For n=10, binomial(10, 5) = 252 = 2*2*3*3*7 has 3 prime divisors of which only one, p=7, is unitary, while 2 and 3 are not. So a(10)=2.

For n=256, binomial(256, 128) also has only 2 prime divisors (3 and 13) whose exponents exceed 1 (4 and 2, respectively), thus a(256)=2.

MATHEMATICA

Table[Count[FactorInteger[Binomial[n, Floor[n/2]]][[All, -1]], e_ /; e > 1], {n, 105}] (* Michael De Vlieger, Mar 05 2017 *)

PROG

(PARI) a(n)=omega(core(binomial(n, n\2), 1)[2]) \\ Charles R Greathouse IV, Mar 09 2017

CROSSREFS

Cf. A001221, A001405, A034444, A034973, A039593, A056057, A056173.

KEYWORD

nonn

AUTHOR

Labos Elemer, Jul 27 2000

EXTENSIONS

Edited by Jon E. Schoenfield, Mar 05 2017

STATUS

approved

A064032

Product of unitary divisors of binomial(n, floor(n/2)).

+10
0

1, 2, 3, 36, 100, 400, 1225, 24010000, 252047376, 4032758016, 2075562447064149770496, 531343986448422341246976, 75186222935463997063888896, 19247673071478783248355557376, 2940278105018015412903875390625, 566574142904620264536665169363475932852029446342410000000000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..16.

FORMULA

a(n) = A061537(A001405(n)). - Amiram Eldar, Jul 22 2024

MATHEMATICA

f[n_] := n^(2^(PrimeNu[n]-1)); Table[f[Binomial[n, Floor[n/2]]], {n, 1, 20}] (* Amiram Eldar, Jul 22 2024 *)

PROG

(PARI) a(n) = apply(x -> x^(2^(omega(x)-1)), binomial(n, n\2)); \\ Amiram Eldar, Jul 22 2024

CROSSREFS

Cf. A001405, A048243, A056173, A061537, A061538.

KEYWORD

nonn

AUTHOR

Labos Elemer, Sep 13 2001

EXTENSIONS

a(15)-a(16) from Amiram Eldar, Jul 22 2024

STATUS

approved

A064033

Product of non-unitary divisors of binomial(n, floor(n/2)) or a(n) = 1 if all divisors are unitary. See A046098.

+10
0

1, 1, 1, 1, 1, 20, 1, 1, 15876, 1016255020032, 1, 728933458176, 8670998958336, 19247673071478783248355557376, 1714723915100625, 752711194884611945703392100000000, 1, 31226235883841773375939805209600000000, 1, 1357651828905889565182743230460164655087616

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Table of n, a(n) for n=1..20.

FORMULA

a(n) = A061538(A001405(n)).

MATHEMATICA

f[n_] := n^((DivisorSigma[0, n] - 2^PrimeNu[n]) / 2); Table[f[Binomial[n, Floor[n/2]]], {n, 1, 20}] (* Amiram Eldar, Jul 22 2024 *)

PROG

(PARI) a(n) = apply(x -> x^((numdiv(x) - 2^omega(x))/2), binomial(n, n\2)); \\ Amiram Eldar, Jul 22 2024

CROSSREFS

Cf. A001405, A048243, A056173, A061537, A061538.

KEYWORD

nonn

AUTHOR

Labos Elemer, Sep 13 2001

EXTENSIONS

a(18)-a(20) from Amiram Eldar, Jul 22 2024

STATUS

approved

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