A060723
a(n) is the denominator of r(n) where r(n) is the sequence of rational numbers defined by the recursion: r(0) = 0, r(1) = 1 and for n>1 r(n) = r(n-1) + r(n-2)/2. From this definition it is clear that a(n) is always a power of 2 (see A060755).
Original entry on oeis.org
1, 1, 1, 2, 1, 4, 4, 8, 1, 16, 16, 32, 8, 64, 64, 128, 8, 256, 256, 512, 128, 1024, 1024, 2048, 256, 4096, 4096, 8192, 2048, 16384, 16384, 32768, 1024, 65536, 65536, 131072, 32768, 262144, 262144, 524288, 65536, 1048576, 1048576, 2097152
Offset: 0
Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
The sequence r(n) begins 0, 1, 1, 3/2, 2, 11/4, 15/4, 41/8, 7, 153/16, 209/16, 571/32, 363/16, 2023/64, 2749/64, 7521/128, 5135/64, ...
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Denominator[RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+a[n-2]/2},a,{n,50}]] (* Harvey P. Dale, Mar 07 2016 *)
Table[Denominator[Simplify[((1/2(1 + Sqrt[3]))^x - (1/2(Sqrt[3] - 1))^x Cos[Pi x])/ Sqrt[3]]], {x, 0, 43}] (* Peter Luschny, Jun 02 2018 *)
A060796
Upper central divisor of n-th primorial.
Original entry on oeis.org
2, 3, 6, 15, 55, 182, 715, 3135, 15015, 81345, 448630, 2733549, 17490603, 114388729, 785147363, 5708795638, 43850489690, 342503171205, 2803419704514, 23622001517543, 201817933409378, 1793779635410490, 16342166369958702, 154171363634898185, 1518410187442699518, 15259831781575946565
Offset: 1
n = 8, q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3135, and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _M. F. Hasler_, Sep 20 2011]
Cf.
A060755,
A000196,
A002110,
A033677,
A060776,
A060777,
A061057,
A060795 (x),
A061060 (y-x),
A182987 (x+y),
A061030,
A061031,
A061032,
A061033,
A200744.
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k = 1; Do[k *= Prime[n]; l = Divisors[k]; x = Length[l]; Print[l[[x/2 + 1]]], {n, 1, 24}] (* Ryan Propper, Jul 25 2005 *)
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A060796(n) = divisors(prod(k=1,n,prime(k)))[2^(n-1)+1] \\ Requires stack size > 2^(n+5). - M. F. Hasler, Sep 20 2011
A060795
Write product of first n primes as x*y with x
Original entry on oeis.org
1, 2, 5, 14, 42, 165, 714, 3094, 14858, 79534, 447051, 2714690, 17395070, 114371070, 783152070, 5708587335, 43848093003, 342444658094, 2803119896185, 23619540863730, 201813981102615, 1793779293633437, 16342050964565645, 154170926013430326, 1518409177581024365
Offset: 1
n = 8: q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3094 and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _Colin Barker_, Oct 22 2010]
2*3*5*7 = 210 = 14*15 with difference of 1, so a(4) = 14.
Cf.
A000196,
A060776,
A060777,
A061057,
A060796 (y),
A061060 (y-x),
A182987 (x+y),
A061030,
A061031,
A061032,
A061033,
A060755,
A033677,
A200743.
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F:= proc(n) local P,N,M;
P:= {seq(ithprime(i),i=1..n)};
N:= floor(sqrt(convert(P,`*`)));
M:= map(convert, combinat:-powerset(P),`*`);
max(select(`<=`,M,N))
end proc:
map(F, [$1..20]); # Robert Israel, Feb 22 2016
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a[n_] := (m = Times @@ Prime[Range[n]] ; dd = Divisors[m]; dd[[Length[dd]/2 // Floor]]); Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Oct 15 2016 *)
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a(n) = my(m=prod(i=1, n, prime(i))); divisors(m)[numdiv(m)\2]; \\ Michel Marcus, Feb 22 2016
A060797
Integer part of square root of n-th primorial, A002110(n).
Original entry on oeis.org
1, 1, 2, 5, 14, 48, 173, 714, 3114, 14936, 80434, 447839, 2724103, 17442771, 114379899, 784149081, 5708691485, 43849291330, 342473913399, 2803269796341, 23620771158594, 201815957246321, 1793779464521955, 16342108667160301, 154171144824008979
Offset: 0
n=8, q(8) = 2*3*5*7*11*13*17*19 = 9699690, a(8)=3114. This is between the 128th and 129th divisors of the 8th primorial: 3094 < A000196(9699690)=3114 < 3135.
(In general, x=A002110(n) always has 2^n divisors, and A000196(x) always lies between the k-th and (k+1)-th divisors of x, where k=ceiling(tau(x)/2) and tau(x) is the number of divisors of x.) - _M. F. Hasler_, Sep 02 2012
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a = {}; Do[b = 1; Do[b = b Prime[x], {x, 1, n}]; AppendTo[a, Floor[b^(1/2)]], {n, 1, 100}]; a (* Artur Jasinski *)
Join[{1},Floor[Sqrt[#]]&/@FoldList[Times,Prime[Range[30]]]] (* Harvey P. Dale, Nov 22 2023 *)
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A060797(n)=sqrtint(prod(k=1, n, prime(k))) \\ M. F. Hasler, Sep 02 2012
A305491
a(n) = numerator(r(n)) where r(n) = (((1/2)*(sqrt(3) + 1))^n - ((1/2)*(sqrt(3) - 1))^n * cos(Pi*n))/sqrt(3).
Original entry on oeis.org
0, 1, 1, 3, 2, 11, 15, 41, 7, 153, 209, 571, 195, 2131, 2911, 7953, 679, 29681, 40545, 110771, 37829, 413403, 564719, 1542841, 263445, 5757961, 7865521, 21489003, 7338631, 80198051, 109552575, 299303201, 12776743, 1117014753, 1525870529, 4168755811, 1423656585
Offset: 0
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Table[Numerator[Simplify[((1/2 (Sqrt[3] + 1))^x - (1/2 (Sqrt[3] - 1))^x Cos[Pi x])/Sqrt[3]]], {x, 0, 36}]
Showing 1-5 of 5 results.
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