Gene Ward Smith - cp's OEIS frontend

A279702 a(n) = floor( exp(gamma) k log log k ) - sigma(k), where gamma is Euler's constant (A001620) and sigma(k) is sum of divisors of k (A000203), the n-th colossally abundant number (A004490).

-5, -6, -9, -18, -26, -34, -123, -107, 3953, 90021, 203866, 678250, 3860926, 62168609, 1022130830, 22777519100, 46323907000, 1499885420000, 47625567000000, 318447820000000, 974228630000000, 36070436000000000

Offset: 2

Gene Ward Smith, Dec 17 2016

sign

By Robin's theorem, if the Riemann hypothesis is true the only negative values this sequence attains are the first eight terms; if it is false, it becomes negative again somewhere farther on. Briggs conjectured, in effect, that this sequence is asymptotic to C k / sqrt(log(k)) for some constant C.

G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.

Cf. A004490, A058209.

A279609 a(n) = floor(H(k) + exp(H(k))*log(H(k))) - sigma(k) where H(k) is the k-th harmonic number Sum_{j=1..k} 1/j and k is the n-th colossally abundant number A004490(n).

Original entry on oeis.org

0, 0, 0, 2, 6, 34, 207, 492, 9051, 143828, 306310, 963859, 5155084, 81053635, 1334916490, 29106956400, 58655156200, 1817551636000, 56466287472000, 376943525488000, 1144451930851200, 41803526752345600

Offset: 2

Gene Ward Smith, Dec 15 2016

nonn

By a theorem of J. C. Lagarias, the Riemann hypothesis is equivalent to the proposition that this sequence never takes a negative value. In fact, by inspection it appears to be monotone increasing; this conjecture implies the Riemann hypothesis but is not in any obvious way implied by it. Stronger conjectures are easy to formulate--for example, if F(n) is the function defined by this sequence, then F(n)/2^n also appears to be monotone increasing.

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

Cf. A004490, A057641.

A259632 a(n) = floor(exp(H_k)*log(H_k)) - sigma(k) where k is the n-th colossally abundant number (Sequence A079526 applied to the colossally abundant numbers (A004490).)

Original entry on oeis.org

-2, -2, -3, -2, 0, 28, 199, 483, 9040, 143814, 306295, 963844, 5155067, 81053615, 1334916470, 29106956400, 58655156000, 1817551640000, 56466287000000, 376943530000000, 1144451930000000, 41803527000000000

Offset: 1

Gene Ward Smith, Dec 17 2016

sign

It follows easily from the work of Lagarias that the Riemann hypothesis is equivalent to this sequence's being nonnegative except for the first four terms.

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

Cf. A004490, A079526.

A217055 Prime numbers which are conductors of elliptic curves.

Original entry on oeis.org

11, 17, 19, 37, 43, 53, 61, 67, 73, 79, 83, 89, 101, 109, 113, 131, 139, 163, 179, 197, 229, 233, 269, 277, 307, 331, 347, 353, 359, 373, 389, 431, 433, 443, 467, 503, 557, 563, 571, 593, 643, 659, 677, 701, 709, 733, 739, 797, 811, 827, 829, 997, 1019, 1051

Offset: 1

Gene Ward Smith, Sep 25 2012

nonn

Taken from the data by Armand Brumer and Oisin McGuinness listing 310716 elliptic curves with prime conductor. Note that for some primes, there is more than one elliptic curve with that conductor.

All primes p of the form p = u^2 + 64 for some integer u are in this sequence, as Setzer (1975) proved that for such primes p that there are exactly two elliptic curves E/Q of conductor p. - Robin Visser, Sep 04 2024

			a(1) = 11, as there are no elliptic curves over Q of conductor less than 11, but there are exactly three elliptic curves over Q of conductor equal to 11, for example E : y^2 + y = x^3 - x^2. - _Robin Visser_, Sep 04 2024

Robin Visser, Table of n, a(n) for n = 1..10000 (taken from the Bennett-Gherga-Rechnitzer database).
M. A. Bennett, A. Gherga, and A. Rechnitzer, Computing elliptic curves over Q, Math. Comp., 88 (2019), no. 317, 1341-1390.
Armand Brumer and Oisin McGuiness, 310716 Elliptic Curves of Prime Conductor
J. E. Cremona, Elliptic Curve Data
B. Setzer, Elliptic Curves of Prime Conductor, J. London Math. Soc. (2) 10 (1975), 367-378.

Cf. A005788, A060564, A363793.

# Uses Cremona's database of elliptic curves (works for all p < 500000)
def is_A217055(p):
    if not Integer(p).is_prime(): return False
    return CremonaDatabase().number_of_curves(p) > 0
print([p for p in range(1, 1000) if is_A217055(p)])  # Robin Visser, Sep 04 2024

A123479 Coefficients of series giving the best rational approximations to sqrt(6).

Original entry on oeis.org

20, 1980, 194040, 19013960, 1863174060, 182572043940, 17890197132080, 1753056746899920, 171781670999060100, 16832850701160989900, 1649447587042777950120, 161629030679491078121880, 15837995559003082877994140, 1551961935751622630965303860

Offset: 1

Gene Ward Smith, Sep 28 2006

The partial sums of the series 5/2 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(6), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [2;2], [2;2,4,2], [2;2,4,2,4,2], [2;2,4,2,4,2,4,2] and so forth.

Sequence of numbers x=a(n) such 4*x+1 and 6*x+1 are both square, and their square roots are A138288(n) and A054320(n). - Paul Cleary, Jun 23 2014

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Cf. A123478, A123480, A029549, A123482.

LinearRecurrence[{99,-99,1},{0,20,1980},{2, 25}] (* Paul Cleary, Jun 23 2014 *)

Vec(-20*x/((x-1)*(x^2-98*x+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

a(n+3) = 99*a(n+2) - 99*a(n+1) + a(n).
a(n) = -5/24 + (( + 2*6^(1/2))/48)*(49 + 20*6^(1/2))^n + ((5 - 2*6^(1/2))/48)*(49 - 20*6^(1/2))^n.
G.f.: -20*x / ((x-1)*(x^2-98*x+1)). - Colin Barker, Jun 23 2014

More terms from Colin Barker, Jun 23 2014

A123482 Coefficients of the series giving the best rational approximations to sqrt(11).

Original entry on oeis.org

60, 23940, 9528120, 3792167880, 1509273288180, 600686976527820, 239071907384784240, 95150018452167599760, 37869468272055319920300, 15071953222259565160679700, 5998599512991034878630600360, 2387427534217209622129818263640, 950190160018936438572789038328420

Offset: 1

Gene Ward Smith, Oct 02 2006

The partial sums of the series 10/3 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(11), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [3;3,6,3], [3;3,6,3,6,3], [3;3,6,3,6,3,6,3] and so forth.

G. C. Greubel, Table of n, a(n) for n = 1..380
Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

Cf. A010468, A041014, A041015.

Cf. A123478, A123479, A123480, A029549.

CoefficientList[Series[-60*x/((x - 1)*(x^2 - 398*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 13 2017 *)

Vec(-60*x/((x-1)*(x^2-398*x+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

a(n+3) = 399*a(n+2) - 399*a(n+1) + a(n).
a(n) = -5/33 + (5/66 + 1/44*11^(1/2))*(199 + 60*11^(1/2))^n + (5/66 - 1/44*11^(1/2))*(199 - 60*11^(1/2))^n.
G.f.: -60*x / ((x-1)*(x^2-398*x+1)). - Colin Barker, Jun 23 2014

More terms from Colin Barker, Jun 23 2014

A123478 Coefficients of series giving the best rational approximations to sqrt(7).

Original entry on oeis.org

48, 12240, 3108960, 789663648, 200571457680, 50944360587120, 12939667017670848, 3286624478127808320, 834789677777445642480, 212033291530993065381648, 53855621259194461161296160, 13679115766543862141903843040, 3474441549080881789582414836048

Offset: 1

Gene Ward Smith, Sep 28 2006

The partial sums of the series 8/3 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(7), which constitute every fourth convergent of the continued fraction. The corresponding continued fractions are [2;1,1,1], [2;1,1,1,4,1,1,1], [2;1,1,1,4,1,1,1,4,1,1,1] and so forth.

Harvey P. Dale, Table of n, a(n) for n = 1..416
Index entries for linear recurrences with constant coefficients, signature (255,-255,1).

Cf. A123479, A123480, A029549, A123482.

LinearRecurrence[{255,-255,1},{48,12240,3108960},30] (* Harvey P. Dale, Nov 20 2016 *)

Vec(-48*x/((x-1)*(x^2-254*x+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

a(n+3) = 255 a(n+2) - 255 a(n+1) + a(n).
a(n) = -4/21 + (2/21+1/28*7^(1/2))*(127+48*7^(1/2))^n + (2/21-1/28*7^(1/2))*(127-48*7^(1/2))^n.
G.f.: -48*x / ((x-1)*(x^2-254*x+1)). - Colin Barker, Jun 23 2014

More terms from Colin Barker, Jun 23 2014

A123480 Coefficients of the series giving the best rational approximations to sqrt(3).

Original entry on oeis.org

4, 60, 840, 11704, 163020, 2270580, 31625104, 440480880, 6135107220, 85451020204, 1190179175640, 16577057438760, 230888624967004, 3215863692099300, 44791203064423200, 623860979209825504, 8689262505873133860, 121025814103014048540, 1685672134936323545704

Offset: 1

Gene Ward Smith, Sep 28 2006

The partial sums of the series 2 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(3), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [1;1,2,1], [1;1,2,1,2,1], [1;1,2,1,2,1,2,1], [1;1,2,1,2,1,2,1,2,1] and so forth.

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A123478, A123479, A029549, A123482. A045899, A076139, A076140, A217855.

CoefficientList[Series[-4*x/((x - 1)*(x^2 - 14*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 13 2017 *)

my(x='x+O('x^50)); Vec(-4*x/((x-1)*(x^2-14*x+1))) \\ G. C. Greubel, Oct 13 2017

a(n+3) = 15*a(n+2) - 15*a(n+1) + a(n).
a(n) = -1/3 + (1/6 + 1/12*3^(1/2))*(7 + 4*3^(1/2))^n + (1/6 - 1/12*3^(1/2))*(7 - 4*3^(1/2))^n.
a(n) = 4*A076139(n) = 2*A217855(n) = 1/2*A045899(n) = 4/3*A076140(n). - Peter Bala, Dec 31 2012
G.f.: -4*x/((x-1)*(x^2-14*x+1)). - Colin Barker, Jan 20 2013
a(n) = A001353(n)*A001353(n+1). - Antonio Alberto Olivares, Apr 06 2020

A118778 Total degree of the classical modular curve X_n(0). Also the degree of the classical modular polynomial.

Original entry on oeis.org

1, 4, 6, 9, 10, 18, 14, 20, 20, 30, 22, 38, 26, 42, 40, 42, 34, 62, 38, 60, 56, 66, 46, 82, 54, 78, 66, 84, 58, 122, 62, 88, 88, 102, 84, 126, 74, 114, 104, 126, 82, 168, 86, 132, 128, 138, 94, 172, 104, 166, 136, 156, 106, 198, 132, 170, 152, 174, 118, 254, 122, 186, 172

Offset: 1

Gene Ward Smith, May 22 2006

This is the total degree of the classical modular curve relating j(z) to j(nz), where j is the j-invariant, or elliptic modular function. If F_n(x, y) = 0 is the equation for the curve (the classical modular equation) then F_n(x, x) is the classical modular polynomial and the sequence is also the sequence of degrees for it. When n is a prime, the degree is 2n.

Serge Lang, ''Elliptic Functions'', Addison-Wesley, 1973.

T. D. Noe, Table of n, a(n) for n = 1..1000

with(numtheory): degx := proc (n) # degree of the classical modular curve X0(n) local a, s; s := 0; for a in divisors(n) do if a^2 > n then s := s + 2*a*phi(igcd(a, n/a))/igcd(a, n/a) fi od; if issqr(n) then s := s+phi(sqrt(n)) fi; s end:

degx[n_] := Module[{s = 0}, Do[ If[ a^2 > n, s = s + 2*a*EulerPhi[ GCD[a, n/a]] / GCD[a, n/a]], {a, Divisors[n]}]; If[ IntegerQ[ Sqrt[n]], s = s + EulerPhi[ Sqrt[n] ] ]; s]; Table[ degx[n], {n, 1, 63}] (* Jean-François Alcover, Jan 29 2013, translated from Maple *)

a(n)=2*sumdiv(n,d,if(d^2>n, my(g=gcd(d,n/d)); d*eulerphi(g)/g)) + if(issquare(n,&n),eulerphi(n)) \\ Charles R Greathouse IV, Jan 29 2013

A122533 Coefficients of the series giving the best rational approximations to 1/e.

Original entry on oeis.org

57, 3667, 525153, 133794291, 53325113593, 30632012923107, 23965268215166337, 24499823488381227043, 31709265214216777648761, 50678828500275334077977523, 98023476146668402679417310817

Offset: 1

Gene Ward Smith, Sep 17 2006

The series giving the best rational approximations to 1/e is 1/e = 1/3 + 2/a(1) - 2/a(2) + 2/a(3) - ... The continued fraction for 1/e is [0;2,1, 2,1,1,4,1,1,6,1,1,8...] and the above best approximations give every third convergent, the convergents deriving from [0;2,1], [0;2,1,2, 1,1], [0;2,1,2,1,1,4,1,1] and so forth are the partial sums of the above infinite series.

G. C. Greubel, Table of n, a(n) for n = 1..200

Cf. A003417, A122523.

I:=[57, 3667, 525153]; [n le 3 select I[n] else ((2*n-3)*(16*n^2 - 3)*Self(n-1) + (2*n+1)*(16*(n-1)^2 -3)*Self(n-2) - (2*n+1)*Self(n-3))/(2*n-3): n in [1..30]]; // G. C. Greubel, Oct 27 2024

RecurrenceTable[{a[n]== ((2*n-3)*(16*n^2 -3)*a[n-1] +(2*n+1)*(16*(n-1)^2 - 3)*a[n-2] -(2*n+1)*a[n-3])/(2*n-3), a[1]==57, a[2]==3667, a[3]==525153}, a, {n,30}] (* G. C. Greubel, Oct 27 2024 *)

@CachedFunction
def a(n): # a = A122533
    if n<4: return (0,57,3667,525153)[n]
    else: return ((2*n-3)*(16*n^2 -3)*a(n-1) +(2*n+1)*(16*(n-1)^2 -3)*a(n-2) -(2*n+1)*a(n-3))/(2*n-3)
[a(n) for n in range(1,31)] # G. C. Greubel, Oct 27 2024

a(n+3) = (16*n^2 + 96*n + 141) * a(n+2) + (2*n+7)*(16*n^2 + 64*n + 61)/(2*n+3) * a(n+1) - (2*n+7)/(2*n+3) * a(n). This recurrence relationship is identical to A122523, for the best approximations to e.

cp's OEIS Frontend

User: Gene Ward Smith

A279702 a(n) = floor( exp(gamma) k log log k ) - sigma(k), where gamma is Euler's constant (A001620) and sigma(k) is sum of divisors of k (A000203), the n-th colossally abundant number (A004490).

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A279609 a(n) = floor(H(k) + exp(H(k))*log(H(k))) - sigma(k) where H(k) is the k-th harmonic number Sum_{j=1..k} 1/j and k is the n-th colossally abundant number A004490(n).

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A259632 a(n) = floor(exp(H_k)*log(H_k)) - sigma(k) where k is the n-th colossally abundant number (Sequence A079526 applied to the colossally abundant numbers (A004490).)

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A217055 Prime numbers which are conductors of elliptic curves.

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A123479 Coefficients of series giving the best rational approximations to sqrt(6).

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A123482 Coefficients of the series giving the best rational approximations to sqrt(11).

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A123478 Coefficients of series giving the best rational approximations to sqrt(7).

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A123480 Coefficients of the series giving the best rational approximations to sqrt(3).

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