A080405 -id:A080405 - cp's OEIS frontend

A081389 Number of non-unitary prime divisors of Catalan numbers, i.e., number of those prime factors whose exponent is greater than one.

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

Offset: 1

Labos Elemer, Mar 27 2003

nonn

			For n=25: Catalan(25) = binomial(50,25)/26 = 4861946401452 =(2*2*3*3*7*7)*29*31*37*41*43*47;
unitary prime divisors: {29,31,37,41,43,47};
non-unitary prime divisors: {2,3,7}, so a(25) = 3.

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

Cf. A000108, A034444, A048105, A056169, A056170, A080405, A081386, A081387, A081388.

Table[Boole[n == 1] + PrimeNu@ # - Count[Transpose[FactorInteger@ #][[2]], 1] &@ CatalanNumber@ n, {n, 105}] (* Michael De Vlieger, Feb 25 2017, after Harvey P. Dale at A056169 *)

catalan(n) = binomial(2*n, n)/(n+1);
nbud(n) = #select(x->x!=1, factor(n)[,2]);
a(n) = nbud(catalan(n)); \\ Michel Marcus, Feb 26 2017

a(n) = A056170(A000108(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)).

A081399 Bigomega of the n-th Catalan number: a(n) = A001222(A000108(n)).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 3, 4, 4, 5, 5, 6, 7, 9, 7, 8, 8, 10, 9, 10, 10, 11, 11, 11, 12, 12, 11, 13, 13, 14, 11, 13, 14, 14, 13, 14, 14, 16, 15, 16, 18, 19, 19, 19, 19, 21, 19, 20, 19, 21, 20, 21, 21, 21, 19, 20, 20, 22, 22, 24, 25, 25, 23, 23, 23, 24, 24, 27, 26, 27, 25, 27, 28, 29, 28

Offset: 0

Labos Elemer, Mar 28 2003

It is easy to show that a(n) is between n/log(n) and 2n/log(n) (for n>n0), cf. [Campbell 1984]. The sequence A137687, roughly the middle of this interval, is a fair approximation for A081399. See A137686 for the (signed) difference of the two sequences.

M. F. Hasler, Table of n, a(n) for n = 0..3000.
Douglas M. Campbell, The Computation of Catalan Numbers, Mathematics Magazine, Vol. 57, No. 4. (Sep., 1984), pp. 195-208.

Cf. A001222, A000108, A022559, A023816, A048621, A081405, A120626, A137686-A137687.

Cf. A080405.

with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(binomial(2*n,n)/(1+n)) fi end: seq(a(n), n=0..75); # Zerinvary Lajos, Apr 11 2008

a[n_] := PrimeOmega[ CatalanNumber[n]]; Table[a[n], {n, 0, 75}] (* Jean-François Alcover, Jul 02 2013 *)

A081399(n)=bigomega(prod(i=2,n,(n+i)/i)) \\ M. F. Hasler, Feb 06 2008

a(n)=A001222[A000108(n)]

Edited and extended by M. F. Hasler, Feb 06 2008

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 *)

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A081389 Number of non-unitary prime divisors of Catalan numbers, i.e., number of those prime factors whose exponent is greater than one.

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

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A081399 Bigomega of the n-th Catalan number: a(n) = A001222(A000108(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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