Noam D. Elkies - cp's OEIS frontend

A135710 Positive integers b such that more than one prime factor p of b attains the maximum of (p-1)*v_p(b) where v_p(b) is the valuation of b at p.

12, 45, 80, 90, 144, 180, 189, 240, 360, 378, 448, 637, 720, 756, 945, 1274, 1344, 1512, 1625, 1728, 1890, 1911, 2025, 2240, 2548, 2673, 3024, 3185, 3250, 3780, 3822, 4032, 4050, 4875, 5096, 5346, 5733, 6048, 6125, 6370, 6400, 6500, 6517, 6720, 7007, 7560, 7644

Offset: 1

Noam D. Elkies, Nov 25 2007

Given b, the number of trailing zeros at the end of the base-b representation of x! is asymptotic to x/M where M is the maximum over p|b of (p-1)*v_p(b).
Usually only one prime p attains the maximum and then the number is v_p(x!)/v_p(b) for all but finitely many x.
But for b=12,45,80,90,..., at least two v_p(x!) must be computed. For example: if b=12 then for x=2006 there are 998 trailing zeros due to v_3 but for x=2007 there are 999 due to v_2.

			For b=90 we have (p-1)*v_p(b) = 1, 4, 4 for p = 2, 3, 5 respectively so the maximum of 4 is attained twice (p=3 and p=5).

Eryk LIPKA, Automaticity of the sequence of the last nonzero digits of n! in a fixed base, Journal de Théorie des Nombres de Bordeaux 31 (2019), 283-291. [See Theorem 3.7 on page 290, and consider the complementary sequence.] - Jean-Paul Allouche and Don Reble, Oct 22 2020.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
J.-P. Allouche, J. Shallit, and R. Yassawi, How to prove that a sequence is not automatic, arXiv:2104.13072 [math.NT], 2021.

Cf. A027868, A011371, A054861.

a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 1, a(n-1)) while (s-> nops(s)<2 or (l->
      l[-2] (p-1)*padic[ordp](k, p),
       s))))([numtheory[factorset](k)[]]) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 23 2020

F[n_] := Module[{f, p, v, vmax}, f = FactorInteger[n]; p = f[[All, 1]]; v = Table[ f[[i, 2]]*(p[[i]]-1), {i, 1, Length[p]}]; vmax = Max[v]; Sum[Boole[v[[i]] == vmax], {i, 1, Length[v]}]]; Reap[For[n = 1, n <= 6400, n++, If[F[n] > 1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 09 2014, translated from PARI *)

{ F(n, f,p,v,vmax)= f=factor(n); p=f[,1]; v=vector(length(p),i,f[i,2]*(p[i]-1)); vmax=vecmax(v); sum(i=1,length(v),v[i]==vmax) } for(n=1,6400,if(F(n)>1,print(n)))

A062693 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 3.

Original entry on oeis.org

1254, 2605, 2774, 3502, 4199, 4669, 4895, 6286, 6671, 7230, 7766, 8005, 9015, 9430, 9654, 10199, 10549, 11005, 11029, 12166, 12270, 12534, 12935, 13317, 14965, 15655, 16151, 16206, 16887, 17958, 18221, 19046, 19726, 20005, 20366

Offset: 0

Noam D. Elkies, Jul 04 2001

nonn

Conjectural, as detailed in the pages from which it is extracted (see the first few links at the web site mentioned for details), but the conjecture is supported by much numerical and theoretical evidence.

A. Dujella, A. S.Janfeda, S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8.
N. D. Elkies, Algorithmic (a.k.a. Computational) Number Theory: Tables, Links, etc.
Fidel Ronquillo Nemenzo, All congruent numbers less than 40000, Proc. Japan Acad. Ser. A Math. Sci., Volume 74, Number 1 (1998), 29-31. See Table IV p. 31.

Cf. A062694, A062695.

r(n)=ellanalyticrank(ellinit([0,0,0,-n^2,0]))[1]
for(n=1,1e4,if(r(n)==3,print1(n", "))) \\ Charles R Greathouse IV, Sep 01 2011

A062695 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 2.

Original entry on oeis.org

34, 41, 65, 137, 138, 145, 154, 161, 194, 210, 219, 226, 257, 265, 291, 299, 313, 323, 330, 353, 371, 386, 395, 410, 426, 434, 442, 457, 465, 505, 514, 546, 561, 602, 609, 651, 658, 674, 689, 721, 723, 731, 761, 777, 793, 866, 889, 890, 905, 915, 985, 987, 995

Offset: 1

Noam D. Elkies, Jul 04 2001

nonn

These n are precisely the primitive congruent numbers (A006991) with n==1, n==2, or n==3 (mod 8). - T. D. Noe, Aug 02 2006

Jinyuan Wang, Table of n, a(n) for n = 1..453
A. Dujella, A. S. Janfeda, S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8
N. D. Elkies, Algorithmic (a.k.a. Computational) Number Theory: Tables, Links, etc.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]
Kazunari Noda and Hideo Wada, All congruent numbers less than 10000, Proc. Japan Acad. Ser. A Math. Sci., Volume 69, Number 6 (1993), 175-178.

Cf. A003273, A006991, A062693, A062694, A259680-A259687.

r(n)=ellanalyticrank(ellinit([0,0,0,-n^2,0]))[1]
for(n=1,1e3,if(issquarefree(n)&&r(n)==2,print1(n", "))) \\ Charles R Greathouse IV, Sep 01 2011; corrected by Frank M Jackson, Aug 04 2016

More terms from Jinyuan Wang, Dec 12 2020

A062694 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 3 and nontrivial SHA[2].

Original entry on oeis.org

42486, 68839, 80189, 82205, 83845, 88502, 92045, 112326, 116645, 125749, 142222, 182005, 199805, 202742, 270805, 275286, 282613, 287246, 295222, 342205, 372742, 392502, 440453, 450079, 473263, 477581, 487302, 488047

Offset: 0

Noam D. Elkies, Jul 04 2001

nonn

Conjectural, as detailed in the pages from which it is extracted (see the first few links at the web site mentioned for details), but the conjecture is supported by much numerical and theoretical evidence.

A. Dujella, A. S.Janfeda, S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8
N. D. Elkies, Algorithmic (a.k.a. Computational) Number Theory: Tables, Links, etc.

Cf. A062693, A062695.

A070030 Number of closed Knight's tours on a 3 X 2n board.

Original entry on oeis.org

0, 0, 0, 0, 16, 176, 1536, 15424, 147728, 1448416, 14060048, 136947616, 1332257856, 12965578752, 126169362176, 1227776129152, 11947846468608, 116266505653888, 1131418872918784, 11010065269439104, 107141489725900544, 1042616896632882688, 10145938076107491328

Offset: 1

Noam D. Elkies, Apr 13 2002

Leonhard Euler stated that no such tours are possible [Memoires Acad. Roy. Sci. (Berlin, 1759), 310-337], and many authors echoed this statement until Ernest Bergholt exhibited solutions for 2n=10 and 12 in British Chess Magazine 1918, page 74. The full set of 16 solutions for 2n=10 was published by Kraitchik in 1927. - Don Knuth, Apr 28 2010
Obtained independently by Don Knuth and Noam D. Elkies in April 1994, both using the transfer-matrix method (in slightly different ways). For this method, see for instance chapter 4.7 of R. P. Stanley's Enumerative Combinatorics, Vol. 1 (1986).
Ian Stewart, Mathematical Recreations column, Scientific American, February 1998, "Feedback", page 95, reports that Richard Ulmer of Denver has sent him a letter reporting work on this subject, about which he is writing a thesis, giving the terms though a(21). - N. J. A. Sloane, Mar 01 2018

			The smallest 3 X 2n board admitting a closed Knight's tour is the 3 X 10, on which there are 16 such tours.

D. E. Knuth, Long and skinny knight's tours, in Selected Papers on Fun and Games, to appear, 2010.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. D. Elkies and R. P. Stanley, The mathematical knight, Math. Intelligencer, 25 (No. 1) (2003), 22-34.
George Jelliss, Open knight's tours of three-rank boards, Knight's Tour Notes, note 3a (21 October 2000).
George Jelliss, Closed knight's tours of three-rank boards, Knight's Tour Notes, note 3b (21 October 2000).
Eric Weisstein's World of Mathematics, Knight's Tour

See also A169764, which gives the number of closed Knight's tours on a 3 X n board. Cf. A169696, A169765-A169777, A001230.

Cf. A158074, A383660.

Rest[CoefficientList[Series[16 * (x^5 + 5*x^6 - 34*x^7 - 116*x^8 + 505*x^9 + 616*x^10 - 3179*x^11 - 4*x^12 + 9536*x^13 - 8176*x^14 - 13392*x^15 + 15360*x^16 + 13888*x^17 + 2784*x^18 - 3328*x^19 - 22016*x^20 + 5120*x^21 + 2048*x^22) / (1 - 6*x - 64*x^2 + 200*x^3 + 1000*x^4 - 3016*x^5 - 3488*x^6 + 24256*x^7 - 23776*x^8 - 104168*x^9 + 203408*x^10 + 184704*x^11 - 443392*x^12 - 14336*x^13 + 151296*x^14 - 145920*x^15 + 263424*x^16 - 317440*x^17 - 36864*x^18 + 966656*x^19 - 573440*x^20 - 131072*x^21), {x, 0, 30}], x]] (* Vaclav Kotesovec, Mar 03 2016 *)

g = 16 * (z^5 + 5*z^6 - 34*z^7 - 116*z^8 + 505*z^9 + 616*z^10 - 3179*z^11 - 4*z^ 12 + 9536*z^13 - 8176*z^14 - 13392*z^15 + 15360*z^16 + 13888*z^17 + 2784*z^18 - 3328*z^19 - 22016*z^20 + 5120*z^21 + 2048*z^22) / (1 - 6*z - 64*z^2 + 200*z^3 + 1000*z^4 - 3016*z^5 - 3488*z^6 + 24256*z^7 - 23776*z^8 - 104168*z^9 + 203408*z^1 0 + 184704*z^11 - 443392*z^12 - 14336*z^13 + 151296*z^14 - 145920*z^15 + 263424* z^16 - 317440*z^17 - 36864*z^18 + 966656*z^19 - 573440*z^20 - 131072*z^21); g = g + O(z^31); vector(30,n,polcoeff(g,n))

Generating function: 16 * (z^5 + 5*z^6 - 34*z^7 - 116*z^8 + 505*z^9 + 616*z^10 - 3179*z^11 - 4*z^12 + 9536*z^13 - 8176*z^14 - 13392*z^15 + 15360*z^16 + 13888*z^17 + 2784*z^18 - 3328*z^19 - 22016*z^20 + 5120*z^21 + 2048*z^22) / (1 - 6*z - 64*z^2 + 200*z^3 + 1000*z^4 - 3016*z^5 - 3488*z^6 + 24256*z^7 - 23776*z^8 - 104168*z^9 + 203408*z^10 + 184704*z^11 - 443392*z^12 - 14336*z^13 + 151296*z^14 - 145920*z^15 + 263424*z^16 - 317440*z^17 - 36864*z^18 + 966656*z^19 - 573440*z^20 - 131072*z^21).
Asymptotic value .0001899*3.11949^(2n). - Don Knuth, Apr 28 2010. More decimal places: a(n) ~ 0.0001898644879979384968655648993009439045986223511152141689341... * 3.1194904567551021585814810124470909514449088706168023079811958...^(2*n). - Vaclav Kotesovec, Mar 03 2016
a(n) = A158074(n)/2. - Eric W. Weisstein, Mar 18 2009
Recurrence (D. E. Knuth and N. D. Elkies, 1994): a(n) = 6*a(n-1) + 64*a(n-2) - 200*a(n-3) - 1000*a(n-4) + 3016*a(n-5) + 3488*a(n-6) - 24256*a(n-7) + 23776*a(n-8) + 104168*a(n-9) - 203408*a(n-10) - 184704*a(n-11) + 443392*a(n-12) + 14336*a(n-13) - 151296*a(n-14) + 145920*a(n-15) - 263424*a(n-16) + 317440*a(n-17) + 36864*a(n-18) - 966656*a(n-19) + 573440*a(n-20) + 131072*a(n-21), for n>=23. - Vaclav Kotesovec, Mar 03 2016
a(n) = A383660(3n). - Don Knuth, May 05 2025

Comment corrected by Johannes W. Meijer, Nov 22 2010

A014619 Exponential generating function is -f(x) * Integral_{t = 0..x} exp(exp(-t) - 1) dt, where f(x) = exp(1 - x - exp(-x)) is the exponential generating function for A014182.

Original entry on oeis.org

-1, 1, 1, -5, 5, 21, -105, 141, 777, -5513, 13209, 39821, -527525, 2257425, -41511, -70561285, 531862173, -1559180499, -8858267353, 147780183829, -936560917615, 1352130196615, 38710924110081, -487251979381019, 2846575686392251, 872653153712201

Offset: 1

Noam D. Elkies

sign

Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
Branko Dragovich, Andrei Yu. Khrennikov, and Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079 [math.NT], 2015.
B. Dragovich and N. Z. Misic, p-Adic invariant summation of some p-adic functional series, P-Adic Numbers, Ultrametric Analysis, and Applications, October 2014, Volume 6, Issue 4, pp 275-283.

Cf. A000670, A040027, A163940.

a[n_] := Sum[(-1)^(n - k + 1) * StirlingS2[n + 1, k + 1] * ((-1)^k * k! * Subfactorial[-k - 1] - Subfactorial[-1]), {k, 0, n}]; Table[a[n] // FullSimplify, {n, 1, 26}] (* Jean-François Alcover, Jan 09 2014, after Vladeta Jovovic *)
nmax = 25; Rest[CoefficientList[Series[E^(-E^(-x) - x) * (Gamma[0, -1] - Gamma[0, -E^(-x)]), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, May 10 2024 *)

a(n)=local(A,B);if(n<0,0, A=exp(-x+x*O(x^n)); B=exp(A-1);n!*polcoeff(-intformal(B)*A/B,n))

E.g.f. A(x) = y satisfies y'' + y'(2-exp(-x)) + y = 0. - Michael Somos, Mar 11 2004
a(n) = Sum_{k = 0..n} (-1)^(n-k+1)*Stirling2(n+1, k+1)*A003422(k). - Vladeta Jovovic, Jan 06 2005
The sequence b(n) = (-1)^n*a(n) satisfies the recurrence: b(n) = -Sum_{i = 1..n} b(i-1)*C(n, i), b(0) = -1. - Ralf Stephan, Feb 24 2005
From Peter Bala, Mar 23 2024: (Start)
It appears that a(n) = Sum_{k = 1..n+1} binomial(n+1, k)*a(k). See Dragovich 2017, Table 1.
If true then the following hold: setting a(0) = -1 then
a(n) = Sum_{k = 1..n-1} (-1)^(n-k)*binomial(n-1, k-1)*a(k-1);
the o.g.f F(x) = - ( x/(1 + x)^2  + x^2/((1 + x)*(1 + 2*x)^2) + x^3/((1 + x)*(1 + 2*x)*(1 + 3*x)^2) + ... ) - Cf. A040027;
F(x) = - x/(1 + x)^2 + x/(1 + x)^2*F(x/(1 + x)). (End)

More terms from Jason Earls, Jun 28 2001

A031507 a(n) = smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 15, 113, 2089, 66265, 1358556

Offset: 0

Noam D. Elkies

See A031508 for the smallest negative k. - Artur Jasinski, Nov 21 2011
See A060950 for the rank of y^2 = x^3 + n. - Jonathan Sondow, Sep 10 2013
Gebel, Pethö, & Zimmer: "One experimental observation derived from the tables is that the rank r of Mordell's curves grows according to r = O(log |k|/|log log |k||^(2/3))." Hence this fit suggests a(n) >> exp(n (log n)^(1/3)) where >> is the Vinogradov symbol. - Charles R Greathouse IV, Sep 10 2013
The curves for k and -27*k are isogenous (as Noam Elkies points out---see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013
Womack (2003) gives further upper bounds: a(7) <= 47550317, a(8) <= 1632201497, a(9) <= 185418133372, a(10) <=  68513487607153. - M. F. Hasler, Jul 01 2024
The three questions for arbitrary k, positive k, and negative k are not very far from each other because the curves for k and -27k are related by a 3-isogeny and therefore have the same rank. It would be most natural to ask for the minimal |k| for k of either sign [see A373795]. - Noam D. Elkies,  Jul 02 2024
a(16) <= 1160221354461565256631205207888 (Elkies, ANTS-XVI, 2024). The same article also establishes the existence of a value of k which has rank >= 17. - N. J. A. Sloane, Jul 05 2024

			a(12) <= 27*A031508(12) <= 27*6533891544658786928 = 176415071705787247056 (from Quer 1987 and Womack). - _Jonathan Sondow_, Sep 10 2013

Noam D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, 2024 Algorithmic Number Theory Symposium, ANTS-XVI, MIT, July 2024.

J. E. Cremona, Elliptic Curve Data
Noam D. Elkies and Zev Klagsbrun, New rank records for elliptic curves having rational torsion, ANTS XIV—Proceedings of the Fourteenth Algorithmic Number Theory Symposium, 233-250. Mathematical Sciences Publishers, Berkeley, CA, 2020.
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367. (doi:10.1023/A:1000281602647 not working as of July 2024.)
J. Quer, Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12, C. R. Acad. Sc. Paris I, 305 (1987), 215-218.
Tom Womack, Explicit Descent on Elliptic Curves, PhD thesis, University of Nottingham, July 2003.
Tom Womack, Minimal-known positive and negative k for Mordell curves of given rank (personal web page, latest available snapshot on web.archive.org from Jan. 2017), last modified Oct. 2002.

Cf. A031508, A373795.

{A031507(n)=for(k=1, oo, ellrank(ellinit([0, k]))[1]==n && return(k))} \\ Use ellanalyticrank() for PARI version < 2.14. - M. F. Hasler, Jul 01 2024

a(n) <= 27*A031508(n) and A031508(n) <= 27*a(n). - Jonathan Sondow, Sep 10 2013

Definition clarified by Jonathan Sondow, Oct 26 2013

Escape clause added to definition by N. J. A. Sloane, Jun 29 2024, because, as John Cremona reminds me, it is not known if k always exists.

A014182 Expansion of e.g.f. exp(1-x-exp(-x)).

Original entry on oeis.org

1, 0, -1, 1, 2, -9, 9, 50, -267, 413, 2180, -17731, 50533, 110176, -1966797, 9938669, -8638718, -278475061, 2540956509, -9816860358, -27172288399, 725503033401, -5592543175252, 15823587507881, 168392610536153, -2848115497132448, 20819319685262839

Offset: 0

Noam D. Elkies

E.g.f. A(x) = y satisfies (y + y' + y'') * y - y'^2 = 0. - Michael Somos, Mar 11 2004
The 10-adic sum: B(n) = Sum_{k>=0} k^n*k! simplifies to: B(n) = A014182(n)*B(0) + A014619(n) for n>=0, where B(0) is the 10-adic sum of factorials (A025016); a result independent of base. - Paul D. Hanna, Aug 12 2006
Equals row sums of triangle A143987 and (shifted) = right border of A143987. [Gary W. Adamson, Sep 07 2008]
From Gary W. Adamson, Dec 31 2008: (Start)
Equals the eigensequence of the inverse of Pascal's triangle, A007318.
Binomial transform shifts to the right: (1, 1, 0, -1, 1, 2, -9, ...).
Double binomial transform = A109747. (End)
Convolved with A154107 = A000110, the Bell numbers. - Gary W. Adamson, Jan 04 2009

			G.f. = 1 - x^2 + x^3 + 2*x^4 - 9*x^5 + 9*x^6 + 50*x^7 - 267*x^8 + 413*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mohammad Ghorbani, Mehdi Hassani, and Hossein Moshtagh, Moments and asymptotic expansion of derangement polynomials in terms of Touchard polynomials, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 4, 832-842. See pp. 834, 839.

Essentially same as A000587. See also A014619.
Cf. A025016.
Cf. A143987, A109747, A154107, A000110.

With[{nn=30},CoefficientList[Series[Exp[1-x-Exp[-x]],{x,0,nn}],x] Range[0,nn]!]  (* Harvey P. Dale, Jan 15 2012 *)
a[ n_] := SeriesCoefficient[ (1 - Sum[ k / Pochhammer[ 1/x + 1, k], {k, n}]) / (1 - x), {x, 0, n} ]; (* Michael Somos, Nov 07 2014 *)

{a(n)=sum(j=0,n,(-1)^(n-j)*Stirling2(n+1,j+1))}
{Stirling2(n,k)=(1/k!)*sum(i=0,k,(-1)^(k-i)*binomial(k,i)*i^n)} \\ Paul D. Hanna, Aug 12 2006

{a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - x - exp( -x + x * O(x^n))), n))} /* Michael Somos, Mar 11 2004 */

def A014182_list(len):  # len>=1
    T = [0]*(len+1); T[1] = 1; R = [1]
    for n in (1..len-1):
        a,b,c = 1,0,0
        for k in range(n,-1,-1):
            r = a - k*b - (k+1)*c
            if k < n : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u; R.append(u)
    return R
A014182_list(27)  # Peter Luschny, Nov 01 2012

E.g.f.: exp(1-x-exp(-x)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n+1,k+1). - Paul D. Hanna, Aug 12 2006
A000587(n+1) = -a(n). - Michael Somos, May 12 2012
G.f.: 1/x/(U(0)-x) -1/x  where U(k)= 1 - x + x*(k+1)/(1 - x/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2012
G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1)/(1 - x/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2012
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+k*x+x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 17 2013
G.f.: G(0)/(1+x)-1 where G(k) = 1 + 1/(1 + k*x - x*(1+k*x)*(1+k*x+x)/(x*(1+k*x+x) + (1+k*x+2*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 09 2013
G.f.: S-1 where S = Sum_{k>=0} (2 + x*k)*x^k/Product_{i=0..k} (1+x+x*i). - Sergei N. Gladkovskii, Feb 09 2013
G.f.: G(0)*x^2/(1+x)/(1+2*x) + 2/(1+x) - 1 where G(k) = 1 + 2/(x + k*x - x^3*(k+1)*(k+2)/(x^2*(k+2) + 2*(1+k*x+3*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 09 2013
G.f.: 1/(x*Q(0)) -1/x, where Q(k) = 1 - x/(1 + (k+1)*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 27 2013
G.f.: G(0)/(1-x)/x - 1/x, where G(k) = 1 - x^2*(k+1)/(x^2*(k+1) + (x*k + 1 - x)*(x*k + 1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 06 2014
G.f.:  (1 - Sum_{k>0} k * x^k / ((1 + x) * (1 + 2*x) + ... (1 + k*x))) / (1 - x). - Michael Somos, Nov 07 2014
a(n) = exp(1) * (-1)^n * Sum_{k>=0} (-1)^k * (k + 1)^n / k!. - Ilya Gutkovskiy, Dec 20 2019

A031508 a(n) = smallest k > 0 such that the elliptic curve y^2 = x^3 - k has rank n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 11, 174, 2351, 28279, 975379

Offset: 0

Noam D. Elkies

See A031507 for the smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n. - Jonathan Sondow, Sep 06 2013
See A060951 for the rank of y^2 = x^3 - n. - Jonathan Sondow, Sep 10 2013
Gebel, Pethö, & Zimmer: "One experimental observation derived from the tables is that the rank r of Mordell's curves grows according to r = O(log |k|/|log log |k||^(2/3))." Hence this fit suggests a(n) >> exp(n (log n)^(1/3)) where >> is the Vinogradov symbol. - Charles R Greathouse IV, Sep 10 2013
a(7) <= 56877643. a(8) <= 2520963512. a(9) <= 463066403167. a(10) <= 56736325657288. a(11) <= 46111487743732324. a(12) <= 6533891544658786928. See Table 3.3 in [Womack 2003]. - Jose Aranda, Jun 30 2024
The three questions for arbitrary k, positive k, and negative k are not very far from each other because the curves for k and -27k are related by a 3-isogeny and therefore have the same rank. It would be most natural to ask for the minimal |k| for k of either sign [see A373795].  - Noam D. Elkies,  Jul 02 2024
a(16) <= 1160221354461565256631205207888 (Elkies, ANTS-XVI, 2024). The same article also establishes the existence of a value of k which has rank >= 17. - N. J. A. Sloane, Jul 05 2024

			From _M. F. Hasler_, Jul 01 2024: (Start)
Sequence A060951 = (0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, ...) gives the analytic rank of the elliptic curve y^2 = x^3 - k for k = 1, 2, 3, ...
We can see that:
  - the smallest k that gives rank 0 is k = 1 = a(0);
  - the smallest k that gives rank 1 is k = 2 = a(1);
  - the smallest k that gives rank 2 is k = 11 = a(2); etc. (End)

Noam D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, 2024 Algorithmic Number Theory Symposium, ANTS-XVI, MIT, July 2024.

J. E. Cremona, Elliptic Curve Data
Noam D. Elkies and Zev Klagsbrun, New rank records for elliptic curves having rational torsion, ANTS XIV—Proceedings of the Fourteenth Algorithmic Number Theory Symposium, 233-250. Mathematical Sciences Publishers, Berkeley, CA, 2020.
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
J. Gebel, A. Pethö, and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (1998), 335-367. MR1602064.
Tom Womack, Explicit Descent on Elliptic Curves, PhD thesis, University of Nottingham, July 2003
Tom Womack, Minimal-known positive and negative k for Mordell curves of given rank (personal web page, latest available snapshot on web.archive.org from Jan. 2017), last modified 10/2002.

Cf. A031507, A373795.

{a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]<>n, k++); k} \\ Seiichi Manyama, Aug 24 2019

{A031508(n)=for(k=1,oo, ellrank(ellinit([0, -k]))[1]==n && return(k))} \\ M. F. Hasler, Jul 01 2024

a(n) = min { k >= 1 | A060951(k) == n }. - M. F. Hasler, Jul 01 2024

Definition clarified by Jonathan Sondow, Oct 26 2013.

Escape clause added to definition by N. J. A. Sloane, Jun 29 2024, because, as John Cremona reminds me, it is not known if k always exists.

cp's OEIS Frontend

User: Noam D. Elkies

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A031507 a(n) = smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.

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