A081389 -id:A081389 - cp's OEIS frontend

A081387 Number of non-unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of prime factors in C(2n,n) whose exponent is greater than one.

0, 0, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 2, 2, 2, 1, 1, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 4, 3, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 2, 4, 2, 2, 3, 3, 3, 3, 4, 4, 5, 4, 4, 2, 3, 3, 2, 2, 2, 4, 3, 3, 4, 3, 4, 5, 4, 2, 2, 2, 3, 5, 5, 5, 5, 3, 2, 3, 2, 3, 3, 3

Offset: 1

Labos Elemer, Mar 27 2003

nonn

			For n = 14: binomial(28,14) = 40116600 = 2*2*2*3*3*3*5*5*17*19*23; unitary prime divisors: {17,19,23}; non-unitary prime divisors: {2,3,5}, so a(14) = 3.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000984, A034444, A048105, A056169, A056170, A067434, A081387, A081388, A081389.

Table[Count[Transpose[FactorInteger[Binomial[2n,n]]][[2]],?(#>1&)],{n,110}] (* _Harvey P. Dale, Oct 08 2012 *)

a(n) = {my(e = factor(binomial(2*n, n))[, 2]); sum(k = 1, #e, e[k] > 1);} \\ Amiram Eldar, Oct 05 2024

a(n) = A056170(A000984(n)) = A001221(A000984(n)) - A081386(n) = A067434(n) - A081386(n).

A081386 Number of unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of those prime factors in C(2n,n), whose exponent equals one.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 3, 4, 4, 4, 5, 5, 4, 3, 5, 7, 6, 7, 7, 8, 9, 9, 6, 7, 7, 7, 8, 11, 12, 11, 11, 11, 12, 12, 12, 13, 13, 13, 11, 13, 12, 14, 13, 13, 15, 14, 14, 14, 15, 16, 16, 16, 17, 19, 18, 17, 18, 19, 18, 19, 18, 18, 18, 20, 18, 21, 22, 20, 20, 20, 20, 20, 20, 19, 21, 21, 24, 23

Offset: 1

Labos Elemer, Mar 27 2003

nonn

			n=10: C(20,10) = 184756 = 2*2*11*13*17*19; unitary-p-divisors = {11,13,17,19}, so a(10)=4.

T. D. Noe, Table of n, a(n) for n=1..2000

Cf. A000984, A067434, A081387-A081389, A034444, A048105, A056170, A056169.

Table[Function[m, Count[Divisors@ m, k_ /; And[PrimeQ@ k, GCD[k, m/k] == 1]]]@ Binomial[2 n, n], {n, 50}] (* Michael De Vlieger, Dec 17 2016 *)

a(n) = my(f=factor(binomial(2*n, n))); sum(k=1, #f~, f[k,2] == 1); \\ Michel Marcus, Dec 18 2016

a(n) = A056169(A000984(n)).

A081388 Number of unitary prime divisors of the n-th Catalan number.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 4, 4, 3, 5, 4, 3, 4, 5, 6, 8, 6, 9, 8, 8, 8, 7, 7, 6, 7, 9, 10, 10, 10, 11, 11, 12, 12, 13, 12, 12, 11, 13, 12, 13, 13, 12, 14, 14, 13, 15, 14, 15, 15, 15, 15, 18, 17, 17, 17, 17, 18, 17, 18, 17, 18, 18, 19, 21, 20, 22, 19, 19, 19, 21, 19, 19, 20, 20, 23, 23, 22

Offset: 1

Labos Elemer, Mar 27 2003

nonn

			For n = 10: Catalan(10) = 16796 = 2^2*13*17*19, the unitary prime divisors are {13, 17, 19}, so a(10) = 3.

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

Cf. A000108, A034444, A048105, A056169, A056170, A080405, A081386, A081387, A081389.

a[n_] := Count[FactorInteger[CatalanNumber[n]][[;;, 2]], 1]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Aug 16 2024 *)

nbupd(n) = my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1);
a(n) = nbupd(binomial(2*n, n)/(n+1)); \\ Michel Marcus, Sep 05 2017

a(n) = A056169(A000108(n)).

A081391 Numbers k such that the central binomial coefficient C(2*k,k) has only one prime divisor whose exponent equals one.

Original entry on oeis.org

3, 6, 7, 8, 9, 10, 11, 12, 16, 21, 22, 28, 29, 30, 31, 36, 37, 54, 55, 57, 58, 110, 171, 784, 786

Offset: 1

Labos Elemer, Mar 27 2003

Numbers k such that C(2*k,k) has one non-unitary prime divisor.
Numbers k for which A081387(k) = 1.
No more terms through 10^6; conjecture: no terms after 786. - Jon E. Schoenfield, Jul 29 2017

			For k = 786, C(1572,786) = 2*2*2*2*m, where m is a squarefree product of 169 primes.

Cf. A000108, A000984, A001405, A046098, A081386, A081387, A081388, A081389, A081390, A080664.

q[k_] := Count[FactorInteger[Binomial[2*k, k]][[;;, 2]], ?( # > 1 &)] == 1; Select[Range[1000], q] (* _Amiram Eldar, Oct 05 2024 *)

is(k) = {my(e = factor(binomial(2*k, k))[, 2]); sum(i = 1, #e, e[i] > 1) == 1;} \\ Amiram Eldar, Oct 05 2024

A081395 a(n) is the smallest value of k such that number of non-unitary prime divisors of k-th Catalan number, A000108(k) = C(2*k,k)/(k+1) equals n.

Original entry on oeis.org

1, 6, 13, 25, 72, 96, 182, 320, 481, 923, 1018, 1321, 1323, 1670, 3457, 3455, 3472, 3464, 3462, 3469, 8222, 9991, 12163, 15838, 17665, 18313, 18480, 19458, 19464, 29708, 36787, 36796, 36789, 40048, 43603, 47210, 47521, 61653, 61675, 80722, 87117, 87120, 92958, 93181, 93186, 93187

Offset: 0

Labos Elemer, Mar 27 2003

nonn

			n=6: a(6)=182, C(364,182)/183 has 6 non-unitary prime divisors: {2,3,7,11,17,19} and 182 is the smallest.

Cf. A000108, A081389, A081393, A081394.

seq[len_, kmax_] := Module[{s = Table[0, {len}], k = 1, c = 0, i}, While[c < len && k < kmax, i = Count[FactorInteger[CatalanNumber[k]][[;; , 2]], ?(# > 1 &)] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = k]; k++]; TakeWhile[s, # > 0 &]]; seq[20, 10^4] (* _Amiram Eldar, May 15 2023 *)

a(n) = Min{k; A081389(k) = n}.

a(11)-a(21) from Michel Marcus, Sep 01 2019

a(22)-a(45) from Amiram Eldar, May 15 2023

A081390 Number k such that the k-th Catalan number has only one non-unitary prime divisor; all the other prime divisors have exponent one.

Original entry on oeis.org

6, 10, 12, 15, 16, 20, 21, 22, 27, 28, 29, 30, 32, 33, 34, 36, 37, 39, 53, 54, 55, 56, 57, 58, 65, 67, 79, 80, 109, 110, 129, 135, 159, 161, 170, 171, 255, 783, 784, 785, 786, 902

Offset: 1

Labos Elemer, Mar 27 2003

a(43) > 25000, if it exists. - Amiram Eldar, Jul 22 2024

			902 is a term because binomial(1804,902)/903 has 189 prime factor, 188 stand with exponent one, but 2 with 5: 2^5.

Cf. A000108, A081386, A081387, A081388, A081389, A080405, A080664.

c[n_] := Count[FactorInteger[n][[;; , 2]], ?(# > 1 &)]; Select[Range[0, 1000], c[CatalanNumber[#]] == 1 &] (* _Amiram Eldar, Jul 22 2024 *)

cp's OEIS Frontend

A081387 Number of non-unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of prime factors in C(2n,n) whose exponent is greater than one.

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A081386 Number of unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of those prime factors in C(2n,n), whose exponent equals one.

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A081388 Number of unitary prime divisors of the n-th Catalan number.

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A081391 Numbers k such that the central binomial coefficient C(2*k,k) has only one prime divisor whose exponent equals one.

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A081395 a(n) is the smallest value of k such that number of non-unitary prime divisors of k-th Catalan number, A000108(k) = C(2*k,k)/(k+1) equals n.

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A081390 Number k such that the k-th Catalan number has only one non-unitary prime divisor; all the other prime divisors have exponent one.

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