Reikku Kulon - cp's OEIS frontend

A307065 Decimal expansion of the negative real attracting fixed point of Э(s) = (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * beta(s) / Pi^s.

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

Offset: 0

Reikku Kulon, Mar 22 2019

Ossicini's function Э(s) is constructed to remove the poles of gamma(s) and zeta(s) along with the trivial zeros of zeta(s) and (Dirichlet) beta(s). Its zeros include the nontrivial zeros of zeta(s) and beta(s), and complex zeros contributed by (1 - 2^s) and (1 - 2^(1 - s)) at regular intervals of 2*Pi/log(2) on the lines Re(s) = {0, 1}.

			-0.1784830971429545702860575466420370769978315915595...

A. Ossicini, An alternative form of the functional equation for Riemann's Zeta function, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008/09), 95-111.

Andrea Ossicini, An Alternative Form of the Functional Equation for Riemann's Zeta Function, II, arXiv:1206.4494 [math.HO], 2012-2014.

Cf. A069857, A069995.

f[s_] := s - (1 - 2^s)(1 - 2^(1-s)) Gamma[s] Zeta[s] ((HurwitzZeta[s, 1/4] - HurwitzZeta[s, 3/4])/(4 Pi)^s);
s0 = s /. FindRoot[f[s], {s, -1/5}, WorkingPrecision -> 100];
RealDigits[s0][[1]] (* Jean-François Alcover, May 07 2019 *)

solve(s = -1/2, -1/8, s - (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * (zetahurwitz(s, 1/4) - zetahurwitz(s, 3/4)) / (4 * Pi)^s)

A324859 Decimal expansion of 0.1990753..., an inflection point of a Hurwitz zeta fixed-point function.

Original entry on oeis.org

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

Offset: 0

Reikku Kulon, Mar 18 2019

For real values of the parameter "a" between 0 and 1, a real fixed point "s" of the iterated Hurwitz zeta function [s = zetahurwitz(s, a)] lies on a curve that passes through A069857 (-0.295905...) and has a maximum tending toward 1. This curve has inflection points for a = 0.1990753... or 0.91964... . The fixed point "s" on this curve for the iteration "s = zetahurwitz(s, A324859)" is A324860 (0.5250984...).

			0.1990753035447728549711300350722284216882866320163...

Reikku Kulon, Plot of Hurwitz zeta fixed-point curve for 0 < a < 2 and -1 < s < +1.

Cf. A069857, A069995, A324860.

solve(t = 1/16, 1/2, derivnum(x = t, solve(v = -1, 1 - x, v - zetahurwitz(v, x)), 2); )

A324860 Decimal expansion of 0.5250984..., a real fixed point of the iteration s = zetahurwitz(s, A324859).

Original entry on oeis.org

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

Offset: 0

Reikku Kulon, Mar 18 2019

For real values of the parameter "a" between 0 and 1, a real fixed point "s" of the iterated Hurwitz zeta function [s = zetahurwitz(s, a)] lies on a curve that passes through A069857 (-0.295905...) and has a maximum tending toward 1. This curve has inflection points for a = 0.1990753... (A324859) or 0.91964... . The fixed point "s" on this curve for the iteration "s = zetahurwitz(s, A324859)" is 0.5250984... .

			0.525098424628892572115438912395851316429631107548...

Cf. A324859, A069857, A069995.

{ A324859 = solve(t = 1/16, 1/2, derivnum(x = t, solve(v = -1, 1 - x, v - zetahurwitz(v, x)), 2); ); solve(v = -1, 1 - A324859, v - zetahurwitz(v, A324859)) }

A285779 Parity index: number of integers z with 1 <= z <= n for which A010060(z) = A010060(n), negated if A010060(n) = 1.

Original entry on oeis.org

0, -1, -2, 1, -3, 2, 3, -4, -5, 4, 5, -6, 6, -7, -8, 7, -9, 8, 9, -10, 10, -11, -12, 11, 12, -13, -14, 13, -15, 14, 15, -16, -17, 16, 17, -18, 18, -19, -20, 19, 20, -21, -22, 21, -23, 22, 23, -24, 24, -25, -26, 25, -27, 26, 27, -28, -29, 28, 29, -30, 30, -31, -32, 31, -33, 32, 33, -34, 34, -35, -36, 35, 36

Offset: 0

Reikku Kulon, Apr 25 2017

Signs are given by A010059 or A010060, the Thue-Morse sequence. Here, zero has positive sign. Like A130472, this sequence maps the natural numbers to the integers. Positive terms are one less than the corresponding term in A008619.
This was a test problem for seqr, a genetic programming integer sequence recognizer, which discovered a method for generating terms of the sequence given the bits of n in descending order.
Iterating over the bits of n in ascending order yields a sequence with more irregular behavior, differing in absolute value by up to 2: 0, -1, -2, 0, -3, +3, +3, +2, -5, 5, 5, ...
Consecutive terms of A285779 usually differ in absolute value by 1 or 2, but consecutive terms differing only in sign occur irregularly. This happens first for a(11) = -6 and a(12) = +6.

Cf. A010059, A010060, A130472, A008619

Function[s, Table[(2 Boole[# == 0] - 1) Count[Take[s, n], z_ /; z == #] &@ s[[n]], {n, 0, Length@ s}]]@ Array[ThueMorse, 72] (* Michael De Vlieger, May 10 2017, Version 10.2 *)

a(n) = {my(v = 1); forstep(b = length(binary(n)) - 1, 0, -1, if(bittest(n, b), v = bitxor(-v, 2^b)); ); v = bitnegimply(v, 1); return(v / 2)}

A190914 Expansion of ( 5-9x^2-2x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).

Original entry on oeis.org

5, 0, 6, 3, 18, 10, 57, 42, 178, 165, 566, 616, 1821, 2236, 5914, 7963, 19362, 27982, 63813, 97394, 211458, 336633, 703786, 1157544, 2350597, 3964960, 7872702, 13541691, 26425522, 46147178, 88853297, 156994354, 299165378, 533410837, 1008343310, 1810544592, 3401446413, 6140811708, 11481472994, 20815538227

Offset: 0

Reikku Kulon, May 23 2011

The sequence ..., 14, 29, 10, 2, 9, 2, 0, [5], 0, 6, 3, 18, 10, 57, 42, ...
(the number in square brackets at index 0) equals the trace of:
  [ 0 0 0 0-1 ]
  [ 1 0 0 0 0 ]
  [ 0 1 0 0 1 ]^(+n)
  [ 0 0 1 0 3 ]
  [ 0 0 0 1 0 ]
or
  [ 0 0 0 0-1 ]
  [ 1 0 0 0 0 ]
  [ 0 1 0 0 3 ]^(-n)
  [ 0 0 1 0 1 ]
  [ 0 0 0 1 0 ]
Its characteristic polynomial is (x^2 +/- x - 1) * (x^3 -/+ x^2 -/+ x - 1); these factors are Fibonacci and tribonacci polynomials.  The ratio of negative terms approaches the golden ratio; the ratio of positive terms approaches the tribonacci constant.
Prime numbers p divide a(+p) and a(-p), as the trace of a matrix M^p (mod p) is constant.
Nonprimes c very rarely divide a(+c) and a(-c) simultaneously.  The only known dual pseudoprime in the sequence is 1.
The distribution of residues induces gaps between pseudoprimes having roughly the size of c.  For example, after 1034881 there is a gap of more than one million terms without either variety of pseudoprime.
Pseudoprimes appear limited to squared primes and squarefree numbers with three or more prime factors.  11 and 13 are more common than other factors.
Positive pseudoprimes: c | a(+c)
----------------------------------------------
1
3481. . . . 59^2
17143 . . . 7 31 79
105589. . . 11 29 331
635335. . . 5 283 449
2992191 . . 3 29 163 211
3659569 . . 1913^2
Negative pseudoprimes: c | a(-c)
----------------------------------------------
1
9 . . . . . 3^2
806 . . . . 2 13 31
1419. . . . 3 11 43
6241. . . . 79^2
6721. . . . 11 13 47
12749 . . . 11 19 61
21106 . . . 2 61 173
38714 . . . 2 13 1489
146689. . . 383^2
649621. . . 7 17 53 103
1034881 . . 41 43 587

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,1,0,-1).

Cf. A190913 (extended to negative indices), A000045, A000073, A001608, A000040, A005117, A125666.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (5-9*x^2 -2*x^3)/((1+x-x^2)*(1-x-x^2-x^3)) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{0, 3, 1, 0, -1}, {5, 0, 6, 3, 18}, 40] (* G. C. Greubel, Apr 23 2019 *)

my(x='x+O('x^40)); Vec((5-9*x^2-2*x^3)/((1+x-x^2)*(1-x-x^2-x^3))) \\ G. C. Greubel, Apr 23 2019

((5-9*x^2-2*x^3)/((1+x-x^2)*(1-x-x^2-x^3))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

a(n) = A061084(n+1) + A001644(n). - R. J. Mathar, Jun 06 2011

A190913 Sequence A190914 evaluated at the negative index -n.

Original entry on oeis.org

5, 0, 2, 9, 2, 10, 29, 14, 50, 99, 82, 220, 365, 416, 926, 1429, 1954, 3842, 5825, 8778, 15922, 24299, 38414, 66240, 102533, 165560, 276954, 434745, 707394, 1163074, 1846069, 3008302, 4900546, 7839115, 12762378, 20694684, 33271421, 54081272, 87516358, 141133157, 229065490, 370410810, 598383689, 970090922, 1568482962

Offset: 0

Reikku Kulon, May 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,3,0,-1)

Cf. A190914, A000045, A000073, A001608, A000040, A005117, A125666.

LinearRecurrence[{0,1,3,0,-1},{5,0,2,9,2},50] (* Vincenzo Librandi, Feb 15 2012 *)

Vec((5-3*x^2-6*x^3)/((x^2+x-1)*(x^3-x^2-x-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 08 2011

From R. J. Mathar, Jun 05 2011: (Start)
a(n) = A190914(-n).
G.f.: ( 5-3*x^2-6*x^3 ) / ( (x^2+x-1)*(x^3-x^2-x-1) ). (End)
a(n) = A000032(n) + A073145(n). - R. J. Mathar, Jun 05 2011

A158036 Integer solutions f for f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) with n an integer.

Original entry on oeis.org

3, 8287, 32547981403, 3374074914839397834392750148706282243018046503, 107547872626305931371847778721098686654377801057464206176785452350259573207, 4568366860875634575966528292411682488942909674818941246717098803707597353756388768388059303363024343431

Offset: 1

Reikku Kulon, Mar 11 2009

nonn

8287 = 129 * 64 + 31 = 257 * 32 + 63 is prime. A158034 (values of n) is often prime. A158035 (2n + 1) appears to be always prime.

See A235540 for nonprimes in A158034. - Reinhard Zumkeller, Nov 17 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..32

Cf. A158034, A158035 (n, 2n + 1)
Cf. A002515 (Lucasian primes)
Cf. A145918 (exponential Sophie Germain primes)
Cf. A235540.

a158036 = (\x -> (4^x - 2^x + 8*x^2 - 2) `div` (2*x*(2*x + 1))) . a158034
-- Reinhard Zumkeller, Nov 17 2014

A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.

Original entry on oeis.org

5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1541, 1559

Offset: 1

Reikku Kulon, Jun 21 2009

Near superset of the Sophie Germain primes (A005384), excluding 2 and 3: 2n + 1 is prime. Nearly all members of this sequence are also prime, but four members less than 10000 are composite: 1541 = 23 * 67, 2465 = 5 * 17 * 29, 3281 = 17 * 193, and 4961 = 11^2 * 41.
The congruence of n modulo 4 is evenly distributed between 1 and 3. n is congruent to 5 (mod 6) for all n less than two billion.
This sequence has roughly twice the density of the sequence (A158034) corresponding to the Diophantine equation
f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)),
and contains most members of that sequence. Those it does not contain are composite and often congruent to 3 (mod 6).
Composite terms appear to predominantly belong to A262051. - Bill McEachen, Aug 29 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A161897 A000040, A002515, A005384, A158034, A158035, A158036, A145918, A002943.

a161896 n = a161896_list !! (n-1)
a161896_list = [x | x <- [1..],
                    (9^x - 3*3^x - 4*x) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

is(n)=my(m=2*n*(2*n+1),t=Mod(3,m)^n); t^2-3*t==4*n \\ Charles R Greathouse IV, Nov 25 2014

A158816 Prime factors of 2^843 - 1.

Original entry on oeis.org

7, 80929, 1390916281, 2475486361, 48009215293052652841860443273079338843737271906291675944391068955229998769420319, 626322472637042112379617556574437460372478130091490111806135154280070248067062629972139895896953692975358576879266688023648639640273675162065398163911

Offset: 1

Reikku Kulon, Mar 27 2009

The 265 bit factor 48009215293052652841860443273079338843737271906291675944391068955229998769420319 was found in less than 30 seconds by Zhenxiang Zhang's method using a 2GHz Athlon. The entire factorization was completed in 47 seconds.

Zhenxiang Zhang, Using Lucas Sequences to Factor Large Integers Near Group Orders, 1999

Cf. A000040, A000225

A160556 Positive integers b for which the Diophantine equation f = (b^(2n) - b^n + 8n^2 - 2) / (2n * (2n + 1)) has at least ten solutions for n <= 10000.

Original entry on oeis.org

2, 8, 14, 17, 26, 29, 32, 38, 41, 47, 50, 59, 62, 64, 65, 68, 74, 77, 83, 89, 95, 98, 101, 104, 110, 119, 122, 128, 131, 134, 137, 140, 143, 149, 152, 155, 161, 164, 167, 173, 179, 182, 185, 188, 194, 197, 200, 206, 209, 212, 215, 218, 221, 224, 227, 230, 233

Offset: 1

Reikku Kulon, May 19 2009

For these equations (not exclusively), the sequences of 2n + 1 are dominated by primes.
When b = 2, there are 105 solutions with n less than 10000, and in this case, the sequence of n is also dominated by primes: only five of these are composite. The average difference between successive composite terms is near the magnitude of n. No composite values of 2n + 1 have been found. n and 2n + 1 account for roughly 3% of primes less than 20 billion. For other bases, n is almost always composite, and 2n + 1 is almost always prime.
The next most productive values of b less than 1000 are 509 (41 solutions) and 824 (40 solutions).
Bases that produce a greater or equal number of solutions than smaller bases, except 2, often have ones digit 4 or 9. Values of n associated with composite 2n + 1 are often divisible by 5.

Cf. A158034, A158035, A158036

Cf. A160557

cp's OEIS Frontend

User: Reikku Kulon

A307065 Decimal expansion of the negative real attracting fixed point of Э(s) = (1 - 2^s) * (1 - 2^(1 - s)) * gamma(s) * zeta(s) * beta(s) / Pi^s.

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A324859 Decimal expansion of 0.1990753..., an inflection point of a Hurwitz zeta fixed-point function.

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A324860 Decimal expansion of 0.5250984..., a real fixed point of the iteration s = zetahurwitz(s, A324859).

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A285779 Parity index: number of integers z with 1 <= z <= n for which A010060(z) = A010060(n), negated if A010060(n) = 1.

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A190914 Expansion of ( 5-9*x^2-2*x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).

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A190913 Sequence A190914 evaluated at the negative index -n.

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Formula

A158036 Integer solutions f for f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) with n an integer.

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A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.

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A190914 Expansion of ( 5-9x^2-2x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).