A165734 -id:A165734 - cp's OEIS frontend

A165949 a(n) = A027762(n)/A165734(n).

1, 1, 7, 1, 11, 91, 1, 17, 133, 11, 23, 91, 1, 29, 2387, 17, 1, 63973, 1, 451, 301, 23, 47, 1547, 11, 53, 133, 29, 59, 1892891, 1, 17, 10787, 1, 781, 4670029, 1, 1, 553, 7667, 83, 113477, 1, 2047, 45353, 47, 1, 150059, 1, 1111, 721, 53, 107, 6973057, 253, 55709, 7

Offset: 1

Paul Curtz, Oct 01 2009

nonn

Cf. A124886.

A165734 := proc(n) op( 1+(n mod 2),[30,6]) ; end:
A027762 := proc(n) a := 0 ; p := 2 ; while p-1 <= 2*n do if (2*n) mod (p-1) = 0 then a := a+1/p ; fi; p := nextprime(p) ; od: denom(a) ; end:
A165949 := proc(n) A027762(n)/A165734(n) ; end: seq(A165949(n),n=1..80) ; # R. J. Mathar, Oct 05 2009

a[n_] := Numerator[ Denominator[ BernoulliB[2*n, 1/2]]/(3*5*2^(2*n))]; Array[a, 57] (* Jean-François Alcover, Apr 17 2013, after Paul Curtz *)

Extended by R. J. Mathar, Oct 05 2009

A176909 Decimal expansion of sqrt(230).

Original entry on oeis.org

1, 5, 1, 6, 5, 7, 5, 0, 8, 8, 8, 1, 0, 3, 1, 0, 1, 1, 0, 8, 5, 1, 3, 6, 5, 0, 8, 7, 2, 5, 6, 4, 1, 4, 3, 1, 0, 9, 0, 9, 9, 2, 8, 4, 2, 7, 9, 0, 3, 4, 9, 2, 4, 5, 0, 6, 4, 5, 2, 4, 2, 0, 0, 3, 5, 5, 8, 4, 4, 6, 6, 7, 7, 3, 6, 5, 0, 8, 4, 9, 9, 3, 1, 8, 2, 8, 1, 2, 1, 8, 6, 2, 8, 2, 7, 8, 9, 2, 8, 0, 7, 0, 6, 7, 3

Offset: 2

Klaus Brockhaus, Apr 28 2010

Continued fraction expansion of sqrt(230) is 15 followed by A165734.

			sqrt(230) = 15.16575088810310110851...

Cf. A002193 (decimal expansion of sqrt(2)), A002163 (decimal expansion of sqrt(5)), A010479 (decimal expansion of sqrt(23)), A176906 (decimal expansion of (15+sqrt(230))/5), A165734 (repeat 6, 30).

A176906 Decimal expansion of (15+sqrt(230))/5.

Original entry on oeis.org

6, 0, 3, 3, 1, 5, 0, 1, 7, 7, 6, 2, 0, 6, 2, 0, 2, 2, 1, 7, 0, 2, 7, 3, 0, 1, 7, 4, 5, 1, 2, 8, 2, 8, 6, 2, 1, 8, 1, 9, 8, 5, 6, 8, 5, 5, 8, 0, 6, 9, 8, 4, 9, 0, 1, 2, 9, 0, 4, 8, 4, 0, 0, 7, 1, 1, 6, 8, 9, 3, 3, 5, 4, 7, 3, 0, 1, 6, 9, 9, 8, 6, 3, 6, 5, 6, 2, 4, 3, 7, 2, 5, 6, 5, 5, 7, 8, 5, 6, 1, 4, 1, 3, 4, 7

Offset: 1

Klaus Brockhaus, Apr 28 2010

Continued fraction expansion of (15+sqrt(230))/5 is A165734.

			(15+sqrt(230))/5 = 6.03315017762062022170...

Cf. A176909 (decimal expansion of sqrt(230)), A165734 (repeat 6, 30).

A216639 A027642(6*n+6)/(sequence of period 2:repeat 42,210).

Original entry on oeis.org

1, 13, 19, 13, 341, 9139, 43, 221, 19, 270413, 1541, 667147, 79, 16211, 6479, 21437, 103, 996151, 1, 11086933, 103759, 20033, 6533, 11341499, 51491, 8545667, 3097, 16211, 59, 34408161359, 1, 4137341, 5826521, 1339, 219666403, 72719023, 223, 2977, 1501, 45423164501, 83

Offset: 0

Paul Curtz, Sep 12 2012

nonn

Is a(n) always an integer?  Is there an a(n) ending with 5?
It appears (tested for n <= 800) that a(n) mod 9 is always one of {1, 2, 4, 5, 7, 8}.
There is a similar sequence of ratios A027642(10n+1)/(66*A010686(n)) which starts 1, 1, 217, 41, 1, 172081, 71, 697, 4123, 101, 23, 7055321, 131, 2059, 32767, 697, 1, 21896102683,...
a(n) is always an integer: 42 = 2*3*7 and 1, 2, and 6 divide 12n+6; 210 = 2*3*5*7 and 1, 2, 4, and 6 divide 12n+12. a(n) never ends in 5 (or 0) since 12n+6 is not divisible by 4 hence the (12n+6)-th Bernoulli denominator is not divisible by 5, and Bernoulli denominators are squarefree and hence the (12n+12)-th Bernoulli denominator, divided by 210, cannot be divisible by 5. - Charles R Greathouse IV, Sep 12 2012
The previous comments argue that 3 or 5 are never prime divisors of a(n). In addition (tested up to n <=900), 7 apparently is also a non-divisor of a(n). In summary, the prime divisors appear all to be in A140461. - Jean-François Alcover, Sep 17 2012

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
A. Joyal, Les nombres de Bernoulli (in French), 2003.

Cf. A002445, A027762, A165734, A165949.

A010686 := proc(n)
    op((n mod 2)+1,[1,5]) ;
end proc:
A216639 := proc(n)
    A027642(6*n+6)/42/A010686(n) ;
end proc: # R. J. Mathar, Sep 21 2012

a(n)=denominator(bernfrac(6*n+6))/if(n%2,210,42) \\ Charles R Greathouse IV, Sep 12 2012

a(n)=my(t=1);fordiv(3*n+3,d,if(isprime(2*d+1),t*=2*d+1));t/if(n%2,105,21) \\ Charles R Greathouse IV, Sep 12 2012

a(n) = A027642(6*n+6)/(42*A010686(n)).

a(20)-a(40) from Charles R Greathouse IV, Sep 12 2012

A228838 a(n) = n * A002445(n).

Original entry on oeis.org

0, 6, 60, 126, 120, 330, 16380, 42, 4080, 7182, 3300, 1518, 32760, 78, 12180, 214830, 8160, 102, 34545420, 114, 270600, 37926, 15180, 6486, 1113840, 1650, 41340, 21546, 24360, 10266, 1703601900, 186, 16320, 2135826, 1020, 164010, 5043631320, 222, 1140

Offset: 0

Paul Curtz, Sep 05 2013

nonn

a(n+1) is a multiple of A040031(n+1), sequence of period 2: 6, 12.
a(n) is divisible by A040879(n)=30 followed by the sequence of period 2: 6, 60. See A040214 and A165734.
Note that A164877(n) + A000367(n) = A164558(2n).

			a(0)=0*1, a(1)=1*6, a(2)=2*30=60,, a(3)=3*42=126.

Colin Barker, Table of n, a(n) for n = 0..1000

PARI
```
a(n)=n*denominator(bernfrac(2*n))
```

a(n) = A176328(2n) - A000367(n).
a(n) = A164877(n)/2.
a(n+1) = A111008(n) * A036283(n+1).
2*a(n) = A164558(2n) - A000367(n).
a(n) = A164558(2n) - A176328(2n).

Typo in data fixed by Colin Barker, Jul 03 2015

A212655 Denominator of Bernoulli(2*n,1/2) / Period of length 2: repeat 12, 60.

Original entry on oeis.org

1, 4, 112, 64, 2816, 93184, 4096, 278528, 8716288, 2883584

Offset: 1

Paul Curtz, Apr 14 2013

See A165949(n) = (A027642(n+1)=A027762(n))/A165734(n).

a(n) is divisible by 4^(n-1).

			a(1) = (B(2,1/2)=12)/12=1, a(2)=240/60=4, a(3)=1344/12=112, a(4)=3840/60=64.

Cf. A000302.

a(n) = A033469(n)/A040874(n).

a(n) = 4^(n-1) * A165949(n).

A214867 Quotients of (first) primorial numbers and denominators of Bernoulli numbers B 0, B 1, B 2, B 4, B 6,... .

Original entry on oeis.org

1, 1, 1, 1, 5, 77, 455, 187, 1616615, 437437, 8107385, 607759061, 53773464745, 111446982977, 2180460221945005, 706769865044243, 2275461421392965, 3770118333635711057, 19548063559901161830545, 4094603218587147211, 92990138354449826827902565

Offset: 0

Paul Curtz, Mar 10 2013

nonn

a(2*n+4) is divisible by 5 (because A006954(n+2)=6,30,42,30,... is divisible by A165734(n)=period of length 2: repeat 6,30).

			a(0) = 1/1, a(1)= 2/2, a(2) = 6/6, a(3) = 30/30, a(4) =210/42=5.
By product (see A080092):
1,
1,
1,
1,
5,
7  * 11,
5  *  7 *13,
11 * 17,
5  *  7 *11 *13 *17 *19,
7  * 11 *13 *19 *23,
5  * 11 *13 *17 *23 *29,
7  * 13 *17 *19 *23 *29 *31,
5  *  7 *11 *13 *17 *19 *29 *31 *37.

a[n_] := Product[ Prime[k], {k,1, n}] / Denominator[ BernoulliB[2*n-2] ]; a[0] = a[1] = 1; Table[a[n],{n,0,20}] (* Jean-François Alcover, Mar 15 2013 *)

a(n) = A002110(n)/A006954(n).

More terms from Jean-François Alcover, Mar 15 2013

cp's OEIS Frontend

A165949 a(n) = A027762(n)/A165734(n).

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A176909 Decimal expansion of sqrt(230).

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A176906 Decimal expansion of (15+sqrt(230))/5.

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A216639 A027642(6*n+6)/(sequence of period 2:repeat 42,210).

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A228838 a(n) = n * A002445(n).

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A212655 Denominator of Bernoulli(2*n,1/2) / Period of length 2: repeat 12, 60.

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A214867 Quotients of (first) primorial numbers and denominators of Bernoulli numbers B 0, B 1, B 2, B 4, B 6,... .

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