A048243 -id:A048243 - cp's OEIS frontend

A064141 Sum of non-unitary divisors of central binomial coefficient C(n, floor(n/2)).

0, 0, 0, 0, 0, 12, 0, 0, 72, 328, 0, 768, 1344, 4032, 3024, 9072, 0, 36288, 0, 120960, 322560, 967680, 0, 1935360, 6013440, 15966720, 43545600, 104094720, 163296000, 362361600, 149299200, 447897600, 1194393600, 4644864000, 2654208000

Offset: 1

Labos Elemer, Sep 11 2001

nonn

			For n = 6, binomial(6,3) = 20 = 4*5, divisors = {1,2,4,5,10,20} of which the non-unitary divisors are 2 and 10 with sum a(6) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from Harry J. Smith)

Cf. A034448, A001405, A046098, A048146, A048243.

nus[n_] := If[n==1, 0, DivisorSigma[1, n] - Times @@ (1 + Power @@@ FactorInteger[n])]; Table[nus@ Binomial[n, Floor[n/2]], {n, 35}] (* Giovanni Resta, Jun 22 2018 *)

usigma(n)= { my(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
a(n)={my(b=binomial(n, n\2)); sigma(b) - usigma(b);} \\ Harry J. Smith, Sep 08 2009

a(n) = A048146(A001405(n)). [corrected by Amiram Eldar, Mar 07 2025]

If n is in A046098 then a(n)=0.

A064146 Sum of non-unitary prime divisors of binomial(n,floor(n/2)).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 3, 5, 0, 2, 2, 2, 3, 3, 0, 2, 0, 2, 2, 2, 0, 2, 7, 7, 10, 10, 5, 5, 3, 3, 3, 5, 5, 7, 7, 7, 3, 5, 2, 2, 2, 2, 10, 10, 8, 10, 12, 12, 12, 12, 9, 9, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 7, 9, 7, 9, 5, 5, 0, 2, 2, 2, 7, 7, 14, 14, 7, 9, 12, 12, 5, 5, 10, 10, 10, 10, 5, 5, 12, 12, 12

Offset: 1

Labos Elemer, Sep 11 2001

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..7500 (first 1000 terms from Harry J. Smith)

Cf. A008472, A001405, A063958, A034444, A056169, A046098, A048243.

a:= n-> add(`if`(i[2]>1, i[1], 0), i=ifactors(binomial(n, iquo(n,2)))[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 24 2018

a[n_] := Sum[If[i[[2]] > 1, i[[1]], 0], {i, FactorInteger[ Binomial[n, Quotient[n, 2]]]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

a(n) = { my(f=factor(binomial(n, n\2))); sum(i=1, #f~, if (f[i, 2]>1, f[i,1])) } \\ Harry J. Smith, Sep 09 2009

a(n) = A063958(A001405(n)).

A064032 Product of unitary divisors of binomial(n, floor(n/2)).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Sep 13 2001

nonn

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

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

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

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

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

A064033 Product of non-unitary divisors of binomial(n, floor(n/2)) or a(n) = 1 if all divisors are unitary. See A046098.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Sep 13 2001

nonn

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

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 *)

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

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

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

cp's OEIS Frontend

A064141 Sum of non-unitary divisors of central binomial coefficient C(n, floor(n/2)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A064146 Sum of non-unitary prime divisors of binomial(n,floor(n/2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A064032 Product of unitary divisors of binomial(n, floor(n/2)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A064033 Product of non-unitary divisors of binomial(n, floor(n/2)) or a(n) = 1 if all divisors are unitary. See A046098.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions