A002445 -id:A002445 - cp's OEIS frontend

A100194 Incrementally largest denominators of the Bernoulli numbers.

1, 2, 6, 30, 42, 66, 2730, 14322, 1919190, 56786730, 140100870, 209191710, 2328255930, 2381714790, 7225713885390, 9538864545210, 21626561658972270, 446617991732222310, 115471236091149548610, 5145485882746933233510, 14493038256293268734790

Offset: 1

Eric W. Weisstein, Nov 08 2004

nonn

Daniel Suteu, Table of n, a(n) for n = 1..85
Eric Weisstein's World of Mathematics, Bernoulli Number
Wikipedia, Von Staudt-Clausen theorem

Cf. A100195, A000367/A002445.

Reap[For[n = record = 0, n < 1000, n = n + 2, If[(d = Denominator[BernoulliB[n]]) > record, Sow[d]; record = d]]][[2, 1]] (* Jean-François Alcover, Nov 09 2012 *)

b(n) = if((n==0) || (n>1 && n%2==1), 1, my(d=divisors(n)); prod(k=1, #d, if(isprime(d[k]+1), d[k]+1, 1))); \\ more efficient than denominator(bernfrac(n))
lista(n) = { my(m=0); for(k=0, n, my(d=b(k)); if(d>m, m=d; print1(d, ", "))); }
lista(1000); \\ Daniel Suteu, Dec 22 2018

a(21) from Seiichi Manyama, Jan 21 2017

A100195 Numbers n such that the denominator of BernoulliB(n) is a record.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 12, 30, 36, 60, 72, 108, 120, 144, 180, 240, 360, 420, 540, 840, 1008, 1080, 1200, 1260, 1620, 1680, 2016, 2160, 2520, 3360, 3780, 5040, 6480, 7560, 8400, 10080, 12600, 15120, 25200, 30240, 42840, 45360, 55440, 60480, 75600, 85680, 100800

Offset: 1

Eric W. Weisstein, Nov 08 2004

nonn

Daniel Suteu, Table of n, a(n) for n = 1..90
Eric Weisstein's World of Mathematics, Bernoulli Number
Wikipedia, Von Staudt-Clausen theorem

Cf. A100194 (the corresponding Bernoulli denominators), A000367/A002445.

DeleteDuplicates[Table[{n,Denominator[BernoulliB[n]]},{n,0,101000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jul 22 2025 *)

b(n) = if((n==0) || (n>1 && n%2==1), 1, my(d=divisors(n)); prod(k=1, #d, if(isprime(d[k]+1), d[k]+1, 1))); \\ more efficient than denominator(bernfrac(n))
lista(n) = { my(m=0); for(k=0, n, my(d=b(k)); if(d>m, m=d; print1(k, ", "))); }
lista(100000); \\ Daniel Suteu, Dec 23 2018

Corrected and extended by Daniel Suteu, Dec 23 2018

A106741 Numbers n such that n divides the denominator of 2n-th Bernoulli number.

Original entry on oeis.org

1, 2, 3, 6, 10, 21, 30, 42, 78, 110, 210, 330, 390, 546, 903, 930, 1218, 1806, 1830, 2310, 2530, 2730, 4134, 4290, 6090, 6162, 6510, 7590, 9030, 10230, 12090, 12246, 12810, 14910, 15834, 20130, 20670, 22110, 23478, 23790, 28938, 30030, 30810, 43134

Offset: 1

Benoit Cloitre, May 15 2005

nonn

Numbers n such that the congruence k^(2n+1) == k (mod n) is true for 1<=k<=n. - Michel Lagneau, May 02 2012

In 2005, B. C. Kellner proved E. W. Weisstein's conjecture that denom(B_n) = n only if n = 1806. - Jonathan Sondow, Oct 14 2013.

Joerg Arndt, Table of n, a(n) for n = 1..594 (terms <= 10^8)
Bernd C. Kellner, The equation denom(B_n) = n has only one solution, preprint 2005.
Victor Miller, Re: Q about a property of Bernoulli denominators, NMBRTHRY list, May 5, 2012
Eric Weisstein's World of Mathematics, Bernoulli Number

Cf. A002445, A014117.

for n from 1 to 10000 do:
    m:=2*n+1: i:=1:
    for k from 1 to n while(k &^ m mod n =k) do: i:=i+1: od:
    if i=n then print(n) fi:
od: # Michel Lagneau, May 02 2012
A106741_list := proc(searchlimit) local isA106741, i;
isA106741 := proc(n)
  numtheory[divisors](2*n);
  map(i->i+1,%);
  select(isprime,%);
  mul(i,i=%) mod n = 0;
  if % then n else NULL fi end:
seq(isA106741(i),i=1..searchlimit) end:
A106741_list(30000); # Peter Luschny, May 04 2012

okQ[n_] := AllTrue[Range[n], PowerMod[#, 2n+1, n] == Mod[#, n]&];
Reap[For[n = 1, n < 50000, n++, If[okQ[n], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jun 11 2019, after Michel Lagneau *)

is_A106741(n)=denominator(bernfrac(2*n))%n==0 \\ Charles R Greathouse IV, May 02 2012

{ for (n=1, 10^6, m = 2*n + 1; for (k=2, n, if ( Mod(k,n)^m != k,  next(2) ); ); print1(n,", "); ); } /* Joerg Arndt, May 04 2012 */

is_A106741(n)={ my(m=2*n+1); for(k=2, n, Mod(k, n)^m - k & return); 1} /* more than twice faster (in PARI 2.4.2) than with "if(...)" */ \\ M. F. Hasler, May 06 2012

Terms a(19)-a(29) from Michel Lagneau, May 02 2012

Terms >= 10230 by Joerg Arndt, May 04 2012

A110936 a(n) = denominator(Bernoulli(prime(n) - 1))/prime(n).

Original entry on oeis.org

1, 2, 6, 6, 6, 210, 30, 42, 6, 30, 462, 51870, 330, 42, 6, 30, 6, 930930, 966, 66, 1919190, 42, 6, 690, 46410, 330, 42, 6, 1919190, 14790, 34314, 66, 30, 1974, 30, 14322, 11430510, 798, 6, 30, 6, 39921071190, 66, 4501770, 870, 1229718, 43725066, 42, 6

Offset: 1

Vladeta Jovovic, Jan 21 2006

			From _Peter Luschny_, Mar 30 2019: (Start)
n = 12 -> prime(12) - 1 = 37 - 1 = 36,
D = divisors(36) \ {36} = {1, 2, 3, 4, 6, 9, 12, 18},
P = {p: (p-1) in D, p prime} = {2, 3, 5, 7, 13, 19},
Product(P) = 51870 = a(n).
.
n = 18 -> prime(18) - 1 = 61 - 1 = 60,
D = divisors(60) \ {60} = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30},
P = {p: (p-1) in D, p prime} = = {2, 3, 5, 7, 11, 13, 31},
Product(P) = 930930 = a(n).
(End)

Cf. A000040, A002445.

a := proc(n) if not isprime(n+1) then return NULL fi;
numtheory[divisors](n) minus {n};
map(i->i+1, %); mul(i, i=select(isprime, %)) end:
seq(a(n), n=1..226); # Peter Luschny, Mar 30 2019

a[n_] := (p = Prime[n]; Denominator[ BernoulliB[p - 1]]/p); Table[a[n], {n, 1, 49}] (* Jean-François Alcover, Dec 13 2012 *)

6 divides a(n) for n >= 3. a(n) is squarefree. - Peter Luschny, Mar 30 2019

A120084 Numerators of expansion for Debye function for n=2: D(2,x).

Original entry on oeis.org

1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373, 0, 154210205991661, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are found under A120085.

This sequence appears to coincide with A120082.

			Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960, ...].

G. C. Greubel, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 998, equ. 27.1.1 for n=1, with a factor (x^2)/2 extracted.
Wolfdieter Lang, Rationals r(n).

Cf. A000367, A002445, A120080, A120081, A120082, A120083, A120085, A120086, A120087.

[Numerator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // G. C. Greubel, May 02 2023

max = 38; Numerator[CoefficientList[Integrate[Normal[Series[(2*(t^2/(Exp[t]-1)))/x^2, {t, 0, max}]], {t, 0, x}], x]] (* Jean-François Alcover, Oct 04 2011 *)
Table[Numerator[2*(n+1)*BernoulliB[n]/(n+2)!], {n,0,50}] (* G. C. Greubel, May 02 2023 *)

[numerator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = numerator(r(n)), with r(n) = [x^n]( 1 - x/3 + Sum_{k >= 1} (B(2*k)/((k+1)*(2*k)!))*x^(2*k) ), |x|<2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = numerator(2*B(n)/((n+2)*n!)), n >= 0. See the comment on the e.g.f. D(2,x) in A120085. - Wolfdieter Lang, Dec 03 2022

A160327 Decimal expansion of (e-1)/(e+1).

Original entry on oeis.org

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

Offset: 0

Harry J. Smith, May 09 2009

			0.462117157260009758502318483643672548730289280330113038552731815838080...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for transcendental numbers

Cf. A016825 (continued fraction), A086403/A079165 (convergents).

Cf. A000367, A002445.

SetDefaultRealField(RealField(100)); (Exp(1) - 1)/(Exp(1) + 1); // G. C. Greubel, Oct 05 2018

RealDigits[(E-1)/(E+1), 10, 100][[1]] (* G. C. Greubel, Oct 05 2018 *)

default(realprecision, 20080); x=tanh(1/2)*10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b160327.txt", n, " ", d));

(exp(1)-1)/(exp(1)+1) \\ Altug Alkan, Oct 05 2018

(e-1)/(e+1) = tanh(1/2).
Equals 2 * Sum_{k>=1} (2^(2*k)-1)*B(2*k)/(2*k)!, where B(2*k) = A000367(k)/A002445(k) are the Bernoulli numbers. - Amiram Eldar, Nov 25 2020
Equals -i * tan(i/2). - Michal Paulovic, Jan 03 2023

A202318 Let (n)_p denote the exponent of prime p in the prime power factorization of n. Then a(n) is defined by the formulas a(1)=1; for n >= 2, (a(n))_2 = (n)_2, (a(n))_3 = (n)_3 and, for p >= 5, (a(n))_p = 1 + ((2n)/(p-1))_p if p-1|2*n, and (a(n))_p = 0 otherwise.

Original entry on oeis.org

1, 10, 21, 20, 11, 2730, 1, 680, 1197, 550, 23, 5460, 1, 290, 7161, 1360, 1, 5757570, 1, 45100, 6321, 230, 47, 185640, 11, 530, 3591, 580, 59, 283933650, 1, 2720, 32361, 10, 781, 840605220, 1, 10, 1659, 1533400, 83, 23830170, 1, 40940, 408177, 470, 1, 36014160, 1, 277750, 2163, 1060, 107, 1882725390

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 16 2011

nonn

a(n)=1 iff n has form 6n+-1 and, if d >= 5 is a divisor of n, then 2*d+1 is not prime. The places of 1's form sequence A045979.

If p is an odd prime and p^n is the side length of the odd leg of a primitive Pythagorean triangle (PPT) it constrains the other leg and hypotenuse to be (p^(2n)-1)/2 and (p^(2n)+1)/2 and the area to be (p^n-1)p^n(p^n+1)/4. Now consider the term (p^n-1)p^n(p^n+1): it must at least be divisible by 24 for all odd primes p because the area of a PPT is divisible by 6 (see A127922 for n=1). a(n) equals the common divisor of the term (p^n-1)p^n(p^n+1)/24 for all odd primes p. - Frank M Jackson, Dec 09 2017

			Let n=6. Since 2*6+1=13 is prime, the max p that should be considered is 13. We have
  (a(6))_2  = (a(6))_3 = 1,
  (a(6))_5  = (12/4)_5 + 1 = 1,
  (a(6))_7  = (12/6)_7 + 1 = 1,
  (a(6))_13 = (12/12)_13 + 1 = 1.
Thus a(6) = 2*3*5*7*13 = 2730.

Frank M Jackson, Table of n, a(n) for n = 1..10000

Cf. A053657, A045979, A002445, A127922.

Table[Numerator[Exp[Re[Limit[Zeta[s] (Zeta[-1]^(s - 1) - Zeta[-(2*n - 1)]^(s - 1)), s -> 1]]]], {n, 1, 54}] (* Mats Granvik, Feb 05 2016 *)
Table[(lst=Table[p=Prime[m+1]; (p^n-1)p^n(p^n+1), {m, 1, 10}]; GCD@@lst/24), {n, 1, 100}] (* Frank M Jackson, Dec 09 2017 *)
a[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; Array[a[2#+1]/(24 a[2#-1]) &, 100] (* using Jean-François Alcover's program A053657 *)(* Frank M Jackson, Dec 16 2017 *)

a(n) = {my(r = 1); forprime(p=2, 2*n+1, if (p<=3, r *= p^valuation(n, p), if (! (2*n % (p-1)), r *= p^(1+valuation((2*n)/(p-1), p))););); r;} \\ Michel Marcus, Feb 06 2016

a(n) = (1/24)*b(2n+1)/b(2n-1), where b(n) = A053657(n).
a(p) = A002445(p)/6, for prime p >= 5.
a(n) = numerator of e^(real(lim_{s -> 1} (zeta(s)*(zeta(-1)^(s-1) - zeta(-(2*n-1))^(s-1))))). - Mats Granvik, Feb 05 2016
a(n) = A036283(n)/6. - Hugo Pfoertner, Dec 18 2022

A300711 a(n) = A000367(n)/A001067(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 25, 13, 1, 7, 29, 1, 31, 1, 11, 17, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 5, 23, 47, 1, 49, 1, 17, 13, 53, 1, 5, 7, 19, 29, 59, 1, 61, 31, 1, 1, 65, 11, 67, 17, 23, 7, 71, 1, 73, 37

Offset: 1

Bernd C. Kellner, Mar 11 2018

nonn

a(n) is the trivial factor of the numerator of Bernoulli(2n) that divides 2n.
The remaining part of the (unsigned) numerator equals a product of powers of irregular primes, or 1 if and only if n = 1, 2, 3, 4, 5, 7.
Alternatively, a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing 2n and p-1 does not divide 2n.

			a(5) = 5, since Bernoulli(10) = 5/66 and Bernoulli(10)/10 = 1/132.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

A111008 equals the first entries and slightly differs, see a(35).

Cf. A000367, A001067, A193267, A195989, A300330, A002445, A075180.

using Nemo
function A300711(n)
    b = bernoulli(n)
    div(numerator(b), numerator(b*QQ(1,n)))
end
[A300711(n) for n in 2:2:148] |> println # Peter Luschny, Mar 11 2018

A300711 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(2*n)[2]; for f in F do if not divides(f[1]-1, 2*n) then
P := P*f[1]^f[2] fi od; P end: seq(A300711(n), n=1..74); # Peter Luschny, Mar 12 2018

Table[Numerator[BernoulliB[n]]/Numerator[BernoulliB[n]/n], {n, 2, 100, 2}]

a(n) = gcd(numerator(bernfrac(2*n)), 2*n) \\ Jianing Song, Apr 05 2021

upto(N)=bernvec(N);forstep(n=2,2*N,2,print1(gcd(numerator(bernfrac(n)), n),", ")) \\ Jeppe Stig Nielsen, Jun 22 2023

a(n) = numerator(Bernoulli(2n))/numerator(Bernoulli(2n)/(2n)).
a(n) * A195989(n) = n. - Peter Luschny, Mar 12 2018
From Jianing Song, Apr 05 2021: (Start)
a(n) = gcd(numerator(Bernoulli(2n)), 2n).
a(n) = A002445(n)*(2n)/A075180(2n-1). (End)

A120085 Denominators of expansion for Debye function for n=2: D(2,x).

Original entry on oeis.org

1, 3, 24, 1, 2160, 1, 120960, 1, 6048000, 1, 287400960, 1, 9153720576000, 1, 597793996800, 1, 96035605585920000, 1, 51090942171709440000, 1, 8831434289681203200000, 1, 169213200472701665280000, 1, 22019713777512667702886400000, 1, 2605883287279605645312000000

Offset: 0

Wolfdieter Lang, Jul 20 2006

Numerators are found under A120084.
D(2,x) := (2/x^2)*Integral_{0..x} t^2/(exp(t)-1) dt is the e.g.f. of 2*B(n)/(n+2), n>=0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). Proof by using the e.g.f. for {k*B(k-1)} (with 0 for k=0) and integrating termwise (allowed for |x| <= r < rho with small enough rho).
See the Abramowitz-Stegun link for the integral and an expansion. - Wolfdieter Lang, Jul 16 2013

			Rationals r(n): [1, -1/3, 1/24, 0, -1/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960,...].

G. C. Greubel, Table of n, a(n) for n = 0..445
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 998, equ. 27.1.1 for n=2, multiplied by 2/x^2.

Cf. A000367/A002445, A027641/A027642, A120080/A120081 (D(3,x) expansion), A120082/A120083 (D(1,x) expansion), A120084, A120086, A120087.

[Denominator(2*(n+1)*Bernoulli(n)/Factorial(n+2)): n in [0..50]]; // G. C. Greubel, May 02 2023

max = 25; Denominator[CoefficientList[Integrate[Normal[Series[(2*(t^2/(Exp[t]-1)))/x^2, {t, 0, max}]], {t, 0, x}], x]](* Jean-François Alcover, Oct 04 2011 *)
Table[Denominator[2*(n+1)*BernoulliB[n]/(n+2)!], {n,0,50}] (* G. C. Greubel, May 02 2023 *)

[denominator(2*(n+1)*bernoulli(n)/factorial(n+2)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = denominator(r(n)), with r(n) = [x^n]( 1 - x/3 + Sum_{k >= 1} (B(2*k)/((k+1)*(2*k)!))*x^(2*k) ), |x|<2*pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator(2*B(n)/((n+2)*n!)), n >= 0. See the comment on the e.g.f. D(2,x) above. - Wolfdieter Lang, Jul 16 2013

A138704 Irregular array read by rows: row n contains the continued fraction terms (in order) for the absolute value of B_{2n}, the (2n)th Bernoulli number.

Original entry on oeis.org

1, 0, 6, 0, 30, 0, 42, 0, 30, 0, 13, 5, 0, 3, 1, 19, 3, 11, 1, 6, 7, 10, 1, 5, 1, 2, 2, 54, 1, 33, 1, 2, 3, 2, 529, 8, 20, 2, 6192, 8, 8, 2, 86580, 3, 1, 19, 3, 11, 1425517, 6, 27298231, 14, 1, 2, 1, 14, 601580873, 1, 9, 15, 2, 7, 6, 15116315767, 10, 1, 5, 1, 2, 2, 429614643061, 6

Offset: 0

Leroy Quet, Mar 26 2008

The number of terms in row n is A138705(n).

			The 12th Bernoulli number is -691/2730. Now 691/2730 has the continued fraction 0 + 1/(3 + 1/(1 + 1/(19 + 1/(3 + 1/11)))). So row 6 is (0,3,1,19,3,11).

Michael De Vlieger, Table of n, a(n) for n = 0..2884

Cf. A138701, A138705, A138706, A000367, A002445.

A138704row := proc(n) local B; B := abs(bernoulli(2*n)) ; numtheory[cfrac](B,20,'quotients') ; end: seq(op(A138704row(n)),n=0..20) ; # R. J. Mathar, Jul 20 2009

Array[ContinuedFraction@ Abs@ BernoulliB[2 #] &, 18, 0] // Flatten (* Michael De Vlieger, Oct 18 2017 *)

More terms from R. J. Mathar, Jul 20 2009

cp's OEIS Frontend

A100194 Incrementally largest denominators of the Bernoulli numbers.

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A100195 Numbers n such that the denominator of BernoulliB(n) is a record.

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Mathematica

PARI

Extensions

A106741 Numbers n such that n divides the denominator of 2n-th Bernoulli number.

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Maple

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A110936 a(n) = denominator(Bernoulli(prime(n) - 1))/prime(n).

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Maple

Mathematica

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A120084 Numerators of expansion for Debye function for n=2: D(2,x).

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Magma

Mathematica

SageMath

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A160327 Decimal expansion of (e-1)/(e+1).

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Magma

Mathematica

PARI

PARI

Formula

A202318 Let (n)_p denote the exponent of prime p in the prime power factorization of n. Then a(n) is defined by the formulas a(1)=1; for n >= 2, (a(n))_2 = (n)_2, (a(n))_3 = (n)_3 and, for p >= 5, (a(n))_p = 1 + ((2n)/(p-1))_p if p-1|2*n, and (a(n))_p = 0 otherwise.

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