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Search: a152762 -id:a152762
Displaying 1-6 of 6 results found. page 1
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A152763 Number of divisors of Catalan number A000108(n). +10
7
1, 1, 2, 2, 4, 8, 12, 8, 16, 16, 24, 32, 48, 72, 192, 96, 192, 256, 576, 512, 768, 768, 1024, 1152, 1152, 1728, 1536, 1536, 4096, 4096, 5120, 2048, 6144, 12288, 12288, 8192, 12288, 12288, 24576, 24576, 36864, 98304, 131072, 147456, 196608, 196608, 368640 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From Jianing Song, Jun 16 2022: (Start)
Conjecture: a(2^k-1) < a(2^k-2) for all k >= 3. Checked up to k = 263. Note that Catalan(2^k-1) is odd and Catalan(2^k-2)/Catalan(2^k-1) = 2^(k-1)/(2^(k+1)-3). Suppose that 2^(k+1)-3 = 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 a(2^k-2)/a(2^k-1) = k * Product_{i=1..r} (e_i-r_i+1)/(e_i+1). This seems unlikely to be less than 1. Actually, it seems that a(2^k-2)/a(2^k-1) tends to infinity as n goes to infinity.
Conjecture: a(2^k-1) != a(2^k) for all k. Checked up to k = 265. Note 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 a(2^k)/a(2^k-1) = 2 * Product_{i=1..r} (e_i+r_i+1)/(e_i+1). This seems unlikely to be equal to 1. Among the numbers k <= 265, the number k for which a(2^k)/a(2^k-1) is closest to 1 is k = 70, where a(2^k)/a(2^k-1) = 104/105. (End)
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
FORMULA
a(n) = A000005(A000108(n)).
MAPLE
A000108 := proc(n) binomial(2*n, n)/(n+1) ; end: A152763 := proc(n) numtheory[tau](A000108(n)) ; end: for n from 0 to 80 do printf("%d, ", A152763(n)) ; od: # R. J. Mathar, Dec 15 2008
MATHEMATICA
DivisorSigma[0, CatalanNumber@Range[0, 40]] (* Vladimir Reshetnikov, Nov 13 2015 *)
PROG
(PARI) vector(100, n, n--; numdiv(binomial(2*n, n)/(n+1))) \\ Altug Alkan, Nov 13 2015
(PARI) val(n, p) = (n - vecsum(digits(n, p)))/(p-1); \\ p-adic valuation of n!
a(n) = my(r=1); forprime(p=2, 2*n, r*=val(2*n, p)-val(n, p)-val(n+1, p)+1); r \\ Jianing Song, Jun 16 2022
CROSSREFS
Cf. A000005, A000108, A152761, A152762.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Dec 14 2008
EXTENSIONS
Extended by R. J. Mathar, Dec 15 2008
STATUS
approved
A152761 Sum of divisors of Catalan number A000108(n). +10
6
1, 1, 3, 6, 24, 96, 336, 672, 3024, 9072, 35280, 120960, 483840, 1874880, 10108800, 20217600, 107827200, 398131200, 1919877120, 6051594240, 24710676480, 86487367680, 339771801600, 1141066967040, 4122564526080, 16784726999040 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..1667
FORMULA
a(n) = sigma(A000108(n)) = A000203(A000108(n)).
a(n) = A152762(n) + A000108(n). - R. J. Mathar, Jan 08 2009
MAPLE
with(numtheory): seq(sigma(binomial(2*n, n)/(n+1)), n = 0 .. 25); # Emeric Deutsch, Jan 10 2009
A000108 := proc(n) binomial(2*n, n)/(n+1) ; end: A000203 := proc(n) numtheory[sigma](n) ; end: A152761 := proc(n) A000203(A000108(n)) ; end: for n from 0 to 30 do printf("%d, ", A152761(n)) ; od: # R. J. Mathar, Jan 08 2009
MATHEMATICA
DivisorSigma[1, CatalanNumber[Range[0, 30]]] (* Harvey P. Dale, Apr 17 2015 *)
CROSSREFS
Cf. A000108, A000203.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Dec 14 2008
EXTENSIONS
Extended by R. J. Mathar and Emeric Deutsch, Jan 08 2009
STATUS
approved
A152765 Smallest prime divisor of Catalan number A000108(n), with a(0) = a(1) = 1. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..65537
FORMULA
a(n) = A020639(A000108(n)). - Michel Marcus, Nov 14 2015
MATHEMATICA
FactorInteger[#][[1, 1]]&/@CatalanNumber[Range[2, 80]] (* Harvey P. Dale, Oct 08 2014 *)
PROG
(PARI) a(n) = if (n<=1, 1, factor(binomial(2*n, n)/(n+1))[1, 1]); \\ Michel Marcus, Nov 14 2015; corrected Jun 13 2022
(PARI) 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
(Magma) [Minimum(PrimeDivisors(Catalan(n))): n in [2..100]]; // Vincenzo Librandi, Jan 04 2017
CROSSREFS
Cf. A000108, A000225, A120275, A120303, A139044, A152761, A152762, A152763.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 15 2008, Jan 03 2009
EXTENSIONS
Terms a(0) = a(1) = 1 prepended and more terms added by Antti Karttunen, Jan 12 2019
STATUS
approved
A152982 Sum of proper divisors of Motzkin number A001006(n). +10
3
0, 0, 1, 3, 4, 11, 21, 1, 37, 173, 1648, 3610, 1, 25125, 139086, 474576, 284493, 984021, 6536394, 24265740, 18678381, 96214041, 277799337, 1282283505, 2077807083, 1899874619, 19252363864, 44221482398, 1967547359, 29743945411, 1265868629 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..200
FORMULA
a(n) = A001065(A001006(n)).
EXAMPLE
a(6)=21 because A001006(6)=51, having as proper divisors 1, 3 and 17. - Emeric Deutsch, Dec 31 2008
MAPLE
with(numtheory): M := proc (n) options operator, arrow: sum(binomial(n, 2*k)*binomial(2*k, k)/(k+1), k = 0 .. n) end proc: seq(sigma(M(n))-M(n), n = 0 .. 30); # Emeric Deutsch, Dec 31 2008
MATHEMATICA
mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; propDivSum[n_] := DivisorSigma[1, n] - n; Table[propDivSum[mot[n]], {n, 0, 30}] (* Amiram Eldar, Nov 26 2019 *)
CROSSREFS
Cf. A001065, A001006, A152762, A152981, A152983.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 20 2008
EXTENSIONS
Extended by Emeric Deutsch, Dec 31 2008
STATUS
approved
A152766 Largest proper divisor of the Catalan number A000108(n). +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
LINKS
Amiram Eldar, Table of n, a(n) for n = 2..1669
FORMULA
a(n) = A032742(A000108(n)). - Amiram Eldar, Dec 01 2019
MATHEMATICA
Divisors[CatalanNumber[#]][[-2]]&/@Range[2, 40] (* Harvey P. Dale, Jun 13 2011 *)
CROSSREFS
Cf. A000108, A152761, A152762, A152763, A152764, A152765.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 15 2008
EXTENSIONS
Edited and extended by N. J. A. Sloane, Dec 19 2008
STATUS
approved
A153788 Number of proper divisors of the Catalan number A000108(n). +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = A032741(A000108(n)) = A152763(n) - 1.
MATHEMATICA
DivisorSigma[0, CatalanNumber@Range[0, 1000]] - 1 (* G. C. Greubel, Aug 28 2016 *)
PROG
(PARI) a(n)=numdiv(binomial(2*n, n)/(n+1)) - 1 \\ Charles R Greathouse IV, Aug 29 2016
CROSSREFS
Cf. A000108, A032741, A152762, A152763, A152988.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 18 2009
STATUS
approved
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Last modified August 29 17:51 EDT 2024. Contains 375518 sequences. (Running on oeis4.)