A152763 -id:A152763 - cp's OEIS frontend

A355041 Numbers k such that A152763(2^k) < A152763(2^k-1).

14, 18, 30, 42, 60, 70, 82, 88, 106, 126, 130, 166, 168, 196, 213, 240, 258, 280, 282, 330

Offset: 1

Jianing Song, Jun 16 2022

Note that Catalan(2^k-1) is odd and that Catalan(2^k)/Catalan(2^k-1) = 2 * (2^(k+1)-1)/(2^k+1). Suppose that (2^(k+1)-1)/(2^k+1) = Product_{i=1..r} (p_i)^(e_i), let r_i be the (p_i)-adic valuation of binomial(2*(2^k-1),2^k-1), then A152763(2^k)/A152763(2^k-1) = 2 * Product_{i=1..r} (e_i+r_i+1)/(e_i+1).

Conjecture: there is no prime in this sequence. Among the primes p <= 257, the prime p for which A152763(2^p)/A152763(2^p-1) is the smallest is p = 193, where A152763(2^p)/A152763(2^p-1) = 143/140.

			14 is a term since A152763(2^14) = 4.457... * 10^721 < A152763(2^14-1) = 4.754... * 10^721. Note that Catalan(2^14)/Catalan(2^14-1) = 2 * 32767/16385, 32767/16385 = (7*31*151)/(5*29*113). We have v(N,5) = v(N,31) = v(N,113) = v(N,151) = 1, v(N,7) = 3, v(N,29) = 2 for N = binomial(2*(2^14-1),2^14-1), so A152763(2^14)/A152763(2^14-1) = 2 * ((3+1+1)/(3+1)) * ((1+1+1)/(1+1)) * ((1+1+1)/(1+1)) * ((1-1+1)/(1+1)) * ((2-1+1)/(2+1)) * ((1-1+1)/(1+1)) = 15/16 < 1.
18 is a term since A152763(2^18) = 1.178... * 10^8888 < A152763(2^18-1) = 2.121... * 10^8888. Note that Catalan(2^18)/Catalan(2^18-1) = 2 * 524287/262145, 524287/262145 = 524287/(5*13*37*109). We have v(N,5) = 5, q(N,13) = 2, v(N,37) = v(N,109) = 1, v(N,524287) = 0 for N = binomial(2*(2^18-1),2^18-1), so A152763(2^18)/A152763(2^18-1) = 2 * ((5-1+1)/(5+1)) * ((2-1+1)/(2+1)) * ((1-1+1)/(1+1)) * ((1-1+1)/(1+1)) * ((0+1+1)/(0+1)) = 5/9 < 1.
Values of A152763(2^k)/A152763(2^k-1) for known terms:
  k = 14: 15/16
  k = 18: 5/9
  k = 30: 9/11
  k = 42: 432/455
  k = 60: 64/81
  k = 70: 104/105
  k = 82: 160/243
  k = 88: 16/21
  k = 106: 38/45
  k = 126: 2275/2673
  k = 130: 3773/6400
  k = 166: 216/287
  k = 168: 27/35
  k = 196: 605/897
  k = 213: 1683/1840
  k = 240: 320/343
  k = 258: 732875/810432

Cf. A152763, A000108, A038003 (the odd Catalan numbers).

val(n,p) = (n - vecsum(digits(n,p)))/(p-1);
q(n,p) = val(2*n,p) - 2*val(n,p);
r(n) = my(list = factor((2^(n+1)-1)/(2^n+1)), w=#list~, rat=2, ex); for(i=1, w, ex=q(2^n-1,list[i,1]); rat*=(ex+list[i,2]+1)/(ex+1)); rat \\ A152763(2^n)/A152763(2^n-1)
isA355041(n) = (r(n) < 1)

a(18)-a(20) from Amiram Eldar, Jul 24 2024

A152762 Sum of proper divisors of Catalan number A000108(n).

Original entry on oeis.org

0, 0, 1, 1, 10, 54, 204, 243, 1594, 4210, 18484, 62174, 275828, 1131980, 7434360, 10522755, 72469530, 268486410, 1442238420, 4284331050, 18146556060, 62021100660, 248289237960, 798007353390, 2832660378756, 11922780597588

Offset: 0

Omar E. Pol, Dec 14 2008

			a(4)=10 because the proper divisors of A000108(4)=14 are 1,2 and 7. - _Emeric Deutsch_, Dec 24 2008

Amiram Eldar, Table of n, a(n) for n = 0..1667

Cf. A000108, A000203, A001065, A152761, A152763.

with(numtheory): seq(sigma(binomial(2*n, n)/(n+1))-binomial(2*n, n)/(n+1), n = 0 .. 27); # Emeric Deutsch, Dec 24 2008

DivisorSigma[1,#]-#&/@CatalanNumber[Range[0,30]] (* Harvey P. Dale, Dec 05 2015 *)

a(n) = A001065(A000108(n)).

a(n) = sigma(binomial(2n,n)/(n+1)) - binomial(2n,n)/(n+1), where sigma(m) is the sum of the divisors of m. - Emeric Deutsch, Dec 24 2008

Extended by Emeric Deutsch, Dec 24 2008

A152765 Smallest prime divisor of Catalan number A000108(n), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Omar E. Pol, Dec 15 2008, Jan 03 2009

nonn

a(n) <> 2 iff n = 2^k - 1 (A000225). In fact for k>1, a(2^k-1): 5, 3, 3, 7, 3, 3, 7, 3, 3, 3, 3, 3, 3, ..., . (A120275) - Robert G. Wilson v, Nov 14 2015

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000108, A000225, A120275, A120303, A139044, A152761, A152762, A152763.

[Minimum(PrimeDivisors(Catalan(n))): n in [2..100]]; // Vincenzo Librandi, Jan 04 2017

FactorInteger[#][[1,1]]&/@CatalanNumber[Range[2,80]] (* Harvey P. Dale, Oct 08 2014 *)

a(n) = if (n<=1, 1, factor(binomial(2*n, n)/(n+1))[1, 1]); \\ Michel Marcus, Nov 14 2015; corrected Jun 13 2022

A152765(n) = if(n<2,1,my(c=binomial(2*n, n)/(n+1)); forprime(p=2, oo, if(!(c%p),return(p)))); \\ Antti Karttunen, Jan 12 2019

a(n) = A020639(A000108(n)). - Michel Marcus, Nov 14 2015

Terms a(0) = a(1) = 1 prepended and more terms added by Antti Karttunen, Jan 12 2019

A152983 Number of divisors of Motzkin number A001006(n).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 2, 4, 4, 6, 8, 2, 8, 24, 18, 4, 16, 8, 12, 16, 24, 48, 72, 12, 8, 6, 16, 8, 16, 8, 12, 4, 16, 64, 12, 2, 8, 8, 8, 8, 24, 96, 96, 6, 24, 72, 48, 24, 32, 128, 96, 16, 8, 8, 8, 16, 128, 60, 192, 6, 32, 32, 96, 8, 96, 512, 36, 24, 16, 24, 384, 24, 96, 144, 48, 64, 64, 32

Offset: 0

Omar E. Pol, Dec 20 2008

nonn

			a(5)=4 because the Motzkin number M(5)=21 has 4 divisors: 1,3,7 and 21. - _Emeric Deutsch_, Jan 14 2009

Amiram Eldar, Table of n, a(n) for n = 0..200

Cf. A000005, A001006, A152763.

with(numtheory): M := proc (n) options operator, arrow: (sum((-1)^j*binomial(n+1, j)*binomial(2*n-3*j, n), j = 0 .. floor((1/3)*n)))/(n+1) end proc: seq(tau(M(n)), n = 0 .. 82); # Emeric Deutsch, Jan 14 2009

mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; Table[DivisorSigma[0, mot[n]], {n, 0, 50}] (* Amiram Eldar, Nov 26 2019 *)

a(n) = A000005(A001006(n)).

Extended by Emeric Deutsch, Jan 14 2009

A152766 Largest proper divisor of the Catalan number A000108(n).

Original entry on oeis.org

1, 1, 7, 21, 66, 143, 715, 2431, 8398, 29393, 104006, 371450, 1337220, 3231615, 17678835, 64822395, 238819350, 883631595, 3282060210, 12233133510, 45741281820, 171529806825, 644952073662, 2430973200726, 9183676536076, 34766775458002, 131873975875180

Offset: 2

Omar E. Pol, Dec 15 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 2..1669

Cf. A000108, A152761, A152762, A152763, A152764, A152765.

Divisors[CatalanNumber[#]][[-2]]&/@Range[2,40] (* Harvey P. Dale, Jun 13 2011 *)

a(n) = A032742(A000108(n)). - Amiram Eldar, Dec 01 2019

Edited and extended by N. J. A. Sloane, Dec 19 2008

A153788 Number of proper divisors of the Catalan number A000108(n).

Original entry on oeis.org

0, 0, 1, 1, 3, 7, 11, 7, 15, 15, 23, 31, 47, 71, 191, 95, 191, 255, 575, 511, 767, 767, 1023, 1151, 1151, 1727, 1535, 1535, 4095, 4095, 5119, 2047, 6143, 12287, 12287, 8191, 12287, 12287, 24575, 24575, 36863, 98303, 131071, 147455

Offset: 0

Omar E. Pol, Jan 18 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A032741, A152762, A152763, A152988.

DivisorSigma[0, CatalanNumber@Range[0, 1000]] - 1 (* G. C. Greubel, Aug 28 2016 *)

a(n)=numdiv(binomial(2*n,n)/(n+1)) - 1 \\ Charles R Greathouse IV, Aug 29 2016

a(n) = A032741(A000108(n)) = A152763(n) - 1.

cp's OEIS Frontend

A355041 Numbers k such that A152763(2^k) < A152763(2^k-1).

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A152762 Sum of proper divisors of Catalan number A000108(n).

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A152765 Smallest prime divisor of Catalan number A000108(n), with a(0) = a(1) = 1.

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A152983 Number of divisors of Motzkin number A001006(n).

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Maple

Mathematica

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A152766 Largest proper divisor of the Catalan number A000108(n).

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Mathematica

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Extensions

A153788 Number of proper divisors of the Catalan number A000108(n).

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Author

Keywords

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Programs

Mathematica

PARI

Formula