Search: a152762 -id:a152762
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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
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0,3
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COMMENTS
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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)
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FORMULA
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MAPLE
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MATHEMATICA
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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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
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0,3
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FORMULA
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MAPLE
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with(numtheory): seq(sigma(binomial(2*n, n)/(n+1)), n = 0 .. 25); # Emeric Deutsch, Jan 10 2009
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MATHEMATICA
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DivisorSigma[1, CatalanNumber[Range[0, 30]]] (* Harvey P. Dale, Apr 17 2015 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A152765
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Smallest prime divisor of Catalan number A000108(n), with a(0) = a(1) = 1.
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+10
3
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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
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OFFSET
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0,3
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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
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LINKS
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FORMULA
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MATHEMATICA
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FactorInteger[#][[1, 1]]&/@CatalanNumber[Range[2, 80]] (* Harvey P. Dale, Oct 08 2014 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Terms a(0) = a(1) = 1 prepended and more terms added by Antti Karttunen, Jan 12 2019
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STATUS
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approved
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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
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0,4
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EXAMPLE
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A152766
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Largest proper divisor of the Catalan number A000108(n).
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+10
2
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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
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OFFSET
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2,3
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LINKS
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FORMULA
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MATHEMATICA
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Divisors[CatalanNumber[#]][[-2]]&/@Range[2, 40] (* Harvey P. Dale, Jun 13 2011 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A153788
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Number of proper divisors of the Catalan number A000108(n).
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+10
1
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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
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OFFSET
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0,5
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LINKS
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FORMULA
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MATHEMATICA
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DivisorSigma[0, CatalanNumber@Range[0, 1000]] - 1 (* G. C. Greubel, Aug 28 2016 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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