Christopher E. Thompson - cp's OEIS frontend

A330274 Largest positive x such that (x,x+n) is the smallest pair of quadratic residues with difference n, modulo any prime.

9, 4, 1, 10, 4, 14, 9, 1, 9, 12, 5, 4, 11, 13, 1, 9, 10, 15, 11, 10, 4, 14, 4, 1, 15, 10, 9, 26, 16, 12, 9, 4, 16, 21, 1, 21, 23, 14, 16, 9, 15, 14, 17, 16, 4, 22, 9, 1, 16, 25, 25, 29, 19, 16, 9, 25, 30, 27, 16, 4, 24, 22, 1, 21, 16, 22, 29, 22, 31, 30, 10

Offset: 1

Christopher E. Thompson, Dec 08 2019

nonn

There is a finite limit for any n. By considering the pairs (1,n+1), (n^2,n^2+n), (n,2n), (4n,5n), (9n,10n) it can be seen that a(n) <= max(9n,n^2).

			If each of the pairs (1,5),(4,8),(6,10),(3,7) are not both quadratic residues, then (10,14) must be. Moreover, if 3 is a quadratic residue but 2,5,7 and 13 are not, then (10,14) is the smallest pair (x,x+4) which are both quadratic residues. Therefore, a(4)=10.

Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag (1981,1994,2004), section F6 "Patterns of quadratic residues".

Christopher E. Thompson, Table of n, a(n) for n = 1..1000
Emma Lehmer, Patterns of power residues, J. Number Theory 17 (1983) 37-46.

A298472 Numbers n such that n and n-1 are both nontrivial binomial coefficients.

Original entry on oeis.org

21, 36, 56, 253, 496, 561, 1771, 2926, 3655, 5985, 26335, 2895621, 2919736, 6471003, 21474181, 48792381, 346700278, 402073903, 1260501229261, 12864662659597529

Offset: 1

Christopher E. Thompson, Jan 19 2018

nonn

Nontrivial here means binomial(r,s) with 2 <= s <= r-2 (or the sequence would be uninteresting).

Blokhuis et al. show that the values given are complete up to 10^30, and conjecture that there are no more.

			binomial(6,3)=20 and binomial(7,2)=binomial(7,5)=21 are the smallest adjacent pair, so a(1)=21.

Aart Blokhuis, Andries Brouwer, Benne de Weger, Binomial collisions and near collisions, INTEGERS, Volume 17, Article A64, 2017 (also available as arXiv:1707.06893 [math.NT]).

Cf. A003015.

nmax = 1000; t = Table[Binomial[n, k], {n, 4, nmax}, {k, 2, Floor[n/2]}] // Flatten // Sort // DeleteDuplicates; Select[Split[t, #2 == #1+1&], Length[#] > 1&][[All, 2]] (* Jean-François Alcover, Feb 20 2018 *)

A278791 Number of complex cubic fields with discriminant >= -10^n.

Original entry on oeis.org

0, 7, 127, 1520, 17041, 182417, 1905514, 19609185, 199884780, 2024660098, 20422230540

Offset: 1

Christopher E. Thompson, Nov 28 2016

Belabas invented an algorithm to identify all cubic fields with a discriminant bounded by X in essentially linear time, and computed the above values up to a(11).

The number of complex cubic fields with discriminant >= -X is asymptotic to X/(4*zeta(3)) = (0.207976...)*X. The second order term was conjectured by Roberts to be a known constant times X^{5/6}, and this was subsequently proved by Bhargava et al.

Henri Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 426 (and Chapter 8 more generally)

Karim Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237.
Manjul Bhargava, Arul Shankar, Jacob Tsimerman, On the Davenport-Heilbronn theorems and second order terms, Invent. math. 193:2 (2013) 439-499.
David P. Roberts, Density of cubic field discriminants, Math. Comp. 70 (2001), 1699-1705.

Cf. A023679, A278790.

A278790 Number of real cubic fields with discriminant <= 10^n.

Original entry on oeis.org

0, 2, 27, 382, 4804, 54600, 592922, 6248290, 64659361, 661448081, 6715824025

Offset: 1

Christopher E. Thompson, Nov 28 2016

Belabas invented an algorithm to identify all cubic fields with a discriminant bounded by X in essentially linear time, and computed the above values up to a(11).

The number of real cubic fields with discriminant <= X is asymptotic to X/(12*zeta(3)) = (0.069325...)*X. The second order term was conjectured by Roberts to be a known constant times X^{5/6}, and this was subsequently proved by Bhargava et al.

Henri Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 426 (and Chapter 8 more generally).

Karim Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237.
Manjul Bhargava, Arul Shankar, Jacob Tsimerman, On the Davenport-Heilbronn theorems and second order terms, Invent. math. 193:2 (2013) 439-499.
David P. Roberts, Density of cubic field discriminants, Math. Comp. 70 (2001), 1699-1705.

Cf. A006832, A278791.

A274471 Numbers missing from A134419 despite satisfying the necessary congruence conditions (see comments).

Original entry on oeis.org

564, 842, 1284, 2306, 2308, 2402, 2459, 3602, 3650, 3803, 6242, 6338, 6779, 7044, 7058, 7319, 7643, 8088, 8354, 8363, 8402, 8543, 8628, 9122, 9168, 9412, 10607, 10826, 10852, 11257, 11378, 11447, 12203, 12436, 12458, 12722, 12984, 13682, 14162, 14388, 14424, 14639

Offset: 1

Christopher E. Thompson, Jun 24 2016

nonn

A134419 consists of those n where x^2 - n*y^2 = n(n-1)(n+1)/3 has integer solutions for x and y. There are easily verified necessary congruence conditions for that to occur:
  (defining x||y to mean x|y and x and y/x are coprime)
  if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);
  if p^e||n with p=5 or 7 (mod 12), then e is even;
  if 3^e||(n+1) with e>0, then e is odd;
  if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.
However, these conditions are not sufficient. This sequence consists of the numbers n satisfying the congruence conditions but for which the Pellian equation has no integer solutions.
If n = k^2*m where m is squarefree, then a necessary (but not sufficient) condition for n to occur in this sequence is that the narrow class group of quadratic forms of discriminant 4*m has more than one class per genus, or equivalently that the narrow class group is not an elementary 2-group.

Christopher E. Thompson, Table of n, a(n) for n = 1..799 [values up to 250000]

Cf. A134419, A274469, A274470.

A274470 Numbers missing from A001033 despite satisfying the necessary congruence conditions (see comments).

Original entry on oeis.org

4, 564, 1284, 2308, 3601, 7044, 7057, 7513, 8628, 9168, 9412, 10561, 10852, 11257, 12436, 13897, 14113, 14388, 14425, 16144, 16692, 16753, 17124, 17257, 17737, 18064, 18433, 18708, 19408, 19428, 20628, 20688, 20752, 20788, 20977, 21073, 23668, 25153, 27193, 28212, 28228

Offset: 1

Christopher E. Thompson, Jun 24 2016

nonn

A001033 consists of those n for which there is a sequence of n consecutive positive odd squares whose sum is a square. For the associated Pellian equation, see A134419. The necessary congruence conditions described in A274471 apply here:
  (defining x||y to mean x|y and x and y/x are coprime)
  if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);
  if p^e||n with p=5 or 7 (mod 12), then e is even;
  if 3^e||(n+1) with e>0, then e is odd;
  if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.
In addition, in order that the Pellian equation has solutions  of the correct parity, one must have:
  if 2^e||n with e>0, then e is even;
  if n is odd, then n=1 (mod 8).
However, these conditions are not sufficient. This sequence consists of the numbers n that satisfy all of the congruence conditions but for which there is no sequence of n consecutive positive odd squares whose sum is a square.
The term 4 is present despite the Pellian equation having a solution with the correct parity, because it leads only to (-1)^2 + 1^2 + 3^2 + 5^2 = 6^2, and the specification of A001033 disallows squares of negative numbers. In every other case the Pellian equation lacks solutions with the right parity. Note however that it may still have solutions with the opposite parity (this can happen only if n=1 mod 8) and so this sequence is not a subsequence of A274471.

Christopher E. Thompson, Table of n, a(n) for n = 1..325 [values up to 250000]

Cf. A001033, A134419, A274471.

A274469 Numbers missing from A001032 despite satisfying the necessary congruence conditions (see comments).

Original entry on oeis.org

25, 842, 2306, 2402, 2459, 3602, 3650, 3803, 6081, 6242, 6338, 6779, 7058, 7319, 7643, 8088, 8354, 8363, 8402, 8543, 8761, 9122, 10607, 10826, 11257, 11378, 11447, 12203, 12458, 12722, 12984, 13273, 13682, 14162, 14424, 14639, 14738, 15362, 15626, 15698, 16475, 16634

Offset: 1

Christopher E. Thompson, Jun 24 2016

nonn

A001032 consists of those n for which there is a sequence of n consecutive positive squares whose sum is a square. For the associated Pellian equation, see A134419. The necessary congruence conditions described in A274471 apply here:
  (defining x||y to mean x|y and x and y/x are coprime)
  if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);
  if p^e||n with p=5 or 7 (mod 12), then e is even;
  if 3^e||(n+1) with e>0, then e is odd;
  if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.
In addition, in order that the Pellian equation has solutions of the correct parity, one must have:
  if 2^e||n with e>0, then e is odd.
However, these conditions are not sufficient. This sequence consists of the numbers n that satisfy all the congruence conditions but for which there is no sequence of n consecutive positive squares whose sum is a square.
The term 25 is present despite the Pellian equation having a solution with the correct parity, because it leads only to 0^2 + 1^2 + ... + 24^2 = 70^2 and the specification of A001032 requires the squares to be strictly positive. (One may wonder whether this is quite natural, compare A185545.) In every other case the Pellian equation lacks solutions with the right parity. Note however that it may still have solutions with the opposite parity (this can happen only if n=1 mod 8) and so this sequence is not a subsequence of A274471.

Christopher E. Thompson, Table of n, a(n) for n = 1..727 [values up to 250000]
H. T. Freitag, G. M. Phillips, On the sum of consecutive squares, Proc. 6th Intl Res. Conf. Fib. Numbers, Pullman, July 18-22, 1994 (1996), p 137.

Cf. A001032, A134419, A274471.

A273096 Number of rotationally inequivalent minimal relations of roots of unity of weight n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 3, 4, 6, 18, 69

Offset: 0

Christopher E. Thompson, May 15 2016

In this context, a relation of weight n is a multiset of n roots of unity which sum to zero, and it is minimal if no proper nonempty sub-multiset sums to zero. Relations are rotationally equivalent if they are obtained by multiplying each element by a common root of unity.

Mann classified the minimal relations up to weight 7, Conway and Jones up to weight 9, and Poonen and Rubinstein up to weight 12.

			Writing e(x) = exp(2*Pi*i*x), then e(1/6)+e(1/5)+e(2/5)+e(3/5)+e(4/5)+e(5/6) = 0 and this is the unique (up to rotation) minimal relation of weight 6.

J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arithmetica 30(3), 229-240 (1976).
Henry B. Mann, On linear relations between roots of unity, Mathematika 12(2), 107-117 (1965).
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Math. 11(1), 135-156 (1998). Also at arXiv:math/9508209 [math.MG] with some typos corrected.

Cf. A103314, A110981, A164896.

A268332 Squarefree numbers differing by more than 3 from any other squarefree number.

Original entry on oeis.org

2526, 44405, 47527, 47973, 55779, 72474, 101037, 106327, 106674, 109023, 110107, 133577, 153173, 165574, 183553, 186247, 193026, 196747, 208847, 209674, 212127, 218527, 220209, 234622, 237522, 245149, 261478, 266853, 269953, 308649, 328877, 334522, 342066, 364151, 370785, 375823

Offset: 1

Christopher E. Thompson, Feb 01 2016

nonn

Christopher E. Thompson, Table of n, a(n) for n = 1..1000

Cf. A005117, A073247, A268331, A268333, A268334.

SF:= select(numtheory:-issqrfree, [$1..10^6]):
SF[select(i -> SF[i]-SF[i-1]>=4  and SF[i+1]-SF[i]>=4, [$2..nops(SF)-1])]; # Robert Israel, Feb 02 2016

A268330 Least squarefree number differing by more than n from any other squarefree number.

Original entry on oeis.org

1, 17, 26, 2526, 5876126, 8061827, 8996188226, 2074150570370

Offset: 0

Christopher E. Thompson, Feb 01 2016

1.8*10^12 < a(7) <= 10735237201449 - Robert Israel, Mar 18 2016

a(8) > 5*10^12. - Giovanni Resta, Apr 11 2016

			a(2) = 26 because 26 is squarefree but 24,25,27,28 are not.

Cf. A073247, A268331, A268332, A268333, A268334.

B = 10^8; % blocks of size B
nB = 1000; % nB blocks
A = [1];
P = primes(floor(sqrt(nB*B)));
mmax = 1;
i0 = 1;
for k = 0:nB-1  % search squarefrees from i0+1 to i0 + B
  V = true(1, B);
  for i = 1:numel(P)
    p = P(i);
    V([(p^2 - mod(i0,p^2)):p^2:B]) = false;
  end
  SF = find(V) + i0;
  DSF = SF(2:end) - SF(1:end-1);
  i0 = SF(end-2);
  M = min(DSF(1:end-1), DSF(2:end));
  newmax = max(mmax,max(M));
  for i = mmax+1:newmax
    A(i) = SF(1 + find(M>=i, 1, 'first'));
  end
  mmax = newmax;
end
for i=1:mmax
  fprintf('%d ',A(i));
end
fprintf('\n');  % Robert Israel, Mar 16 2016

(* implementation assumes a(n) is increasing *)
nsfRun[n_]:=Module[{i=n}, While[!SquareFreeQ[i], i++]; i-n]
a268330[{low_, high_}, width_]:=Module[{k=width, i, next, r, s, list={}}, For[i=low, i<=high, i+=next, r=nsfRun[i]; If[r0 (* Hartmut F. W. Hoft, Mar 15 2016 *)
a268330[{0,10000000},1] (* computes a(1)...a(5) *)

a(6) from Hartmut F. W. Hoft, Mar 15 2016

a(7) from Giovanni Resta, Apr 11 2016

cp's OEIS Frontend

User: Christopher E. Thompson

A330274 Largest positive x such that (x,x+n) is the smallest pair of quadratic residues with difference n, modulo any prime.

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A298472 Numbers n such that n and n-1 are both nontrivial binomial coefficients.

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Mathematica

A278791 Number of complex cubic fields with discriminant >= -10^n.

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A278790 Number of real cubic fields with discriminant <= 10^n.

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A274471 Numbers missing from A134419 despite satisfying the necessary congruence conditions (see comments).

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A274470 Numbers missing from A001033 despite satisfying the necessary congruence conditions (see comments).

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A274469 Numbers missing from A001032 despite satisfying the necessary congruence conditions (see comments).

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A273096 Number of rotationally inequivalent minimal relations of roots of unity of weight n.

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A268332 Squarefree numbers differing by more than 3 from any other squarefree number.

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Maple

A268330 Least squarefree number differing by more than n from any other squarefree number.

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