A328910 -id:A328910 - cp's OEIS frontend

A328980 Erroneous version of A328910: Number of solutions to Erdős's Last Equation in n variables.

3, 7, 8, 8, 17, 14, 18

Offset: 2

N. J. A. Sloane, Jun 04 2020

dead

The equation is x_1*x_2*...*x_n = n*(x_1+...+x_n) (n >= 2).
These are the terms are given as f(n) in a table on p. 803 in the cited paper. However, the sets C_L(n) of solutions with L+1 components > 1 can't be nonempty for L = n, as listed there for n = 2 and n = 3. The author (pers. comm.) confirms that a(2) = 2, a(3) = 6 and a(8) = 19. Sequence A328910 gives the correct terms. - M. F. Hasler, Nov 07 2019
Included in accordance with OEIS policy of including published but erroneous sequences to serve as pointers to the correct values.

Peter Shiu, On Erdős's Last Equation, Amer. Math. Monthly, 126 (2019), 802-808; correction, 127:5 (2020), 478.

Cf. A328910, A328911.

A213820 Principal diagonal of the convolution array A213819.

Original entry on oeis.org

2, 18, 60, 140, 270, 462, 728, 1080, 1530, 2090, 2772, 3588, 4550, 5670, 6960, 8432, 10098, 11970, 14060, 16380, 18942, 21758, 24840, 28200, 31850, 35802, 40068, 44660, 49590, 54870, 60512, 66528, 72930, 79730, 86940, 94572, 102638, 111150, 120120, 129560, 139482

Offset: 1

Clark Kimberling, Jul 04 2012

Every term is even: a(n) = 2*A002414(n).

a(n) is the first Zagreb index of the graph obtained by joining one vertex of a complete graph K[n] with each vertex of a second complete graph K[n]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 07 2016

Clark Kimberling, Table of n, a(n) for n = 1..1000
Ivan Gutman and Kinkar C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002414, A213819, A328910.

(See A213819.)
a[n_] := 2*n^3 + n^2 - n; Array[a, 50] (* Amiram Eldar, Mar 12 2023 *)

a(n) = -n + n^2 + 2*n^3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: f(x)/g(x), where f(x) = 2*x*(1 + 5*x) and g(x) = (1-x)^4.
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = (4*log(2) - 1)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi - 4*log(2) + 1)/3. (End)

A328911 Irregular triangle read by rows: T(n,k) = number of solutions to Erdös's Last Equation x_1...x_n = n(x_1+...+x_n), 0 < x_1 <= ... <= x_n, having k+1 components x_i > 1, 1 <= k <= 2log_2(n).

Original entry on oeis.org

2, 0, 3, 3, 0, 4, 3, 1, 0, 4, 4, 0, 0, 6, 7, 4, 0, 0, 6, 5, 3, 0, 0, 5, 7, 4, 2, 1, 0, 8, 13, 5, 1, 0, 0, 9, 12, 3, 1, 0, 0, 6, 6, 3, 0, 0, 0, 8, 13, 9, 3, 0, 0, 0, 8, 7, 1, 0, 0, 0, 0, 6, 15, 6, 2, 1, 0, 0, 12, 16, 12, 3, 0, 0, 0, 12, 15, 11, 4, 2, 1, 0, 0, 6, 8, 2, 2, 0, 0, 0, 0

Offset: 2

M. F. Hasler, Nov 07 2019

For n = 1 the equation is trivially solved by any integer, therefore we only consider n >= 2.
If any x_k = 0, then all x_i must be zero, so (0, ..., 0) would be the only additional solution in nonnegative integers. This solution is not considered here.
A vector (1, ..., 1, x_n) can never be a solution for n > 1. The number of components different from 1 must be k+1 >= 2 <=> k >= 1.
It can be shown that no solution can have 2^k > n^2, cf. the Shiu paper. Therefore row lengths are floor(2 log_2(n)) = (2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, ...) = A329202(n), n >= 2.
Row sums yield the total number of nontrivial solutions A328910(n), see there for more information.
T(n,k) is equal to |C_k(n)| in the Shiu paper, but some values given in the table on top of p. 803 are erroneous (pers. comm. from the author).

			The table starts:
   n : T(n,k), 1 <= k <= 2*log_2(n)
   2 :   2   0
   3 :   3   3   0
   4 :   4   3   1   0
   5 :   4   4   0   0
   6 :   6   7   4   0   0
   7 :   6   5   3   0   0
   8 :   5   7   4   2   1   0
   9 :   8  13   5   1   0   0
  10 :   9  12   3   1   0   0
  11 :   6   6   3   0   0   0
  12 :   8  13   9   3   0   0   0
  13 :   8   7   1   0   0   0   0
  14 :   6  15   6   2   1   0   0
  15 :  12  16  12   3   0   0   0
For n = 2 variables, we have the equation x1*x2 = 2*(x1 + x2) with positive integer solutions (3,6) and (4,4): Both have k+1 = 2 components > 1, i.e., k = 1.
For n = 3, we have T(3,1) = 3 solutions with k+1 = 2 components > 1, {(1, 4, 15), (1, 5, 9), (1, 6, 7)}, and T(3,2) = 3 with k+1 = 3 components > 1, {(2, 2, 12), (2, 3, 5), (3,3,3)}.
For n = 4 we have the 8 solutions (1, 1, 5, 28), (1, 1, 6, 16), (1, 1, 7, 12), (1, 1, 8, 10), (1, 2, 3, 12), (1, 2, 4, 7), (1, 3, 4, 4) and (2, 2, 2, 6). Four of them have k+1 = 2 components > 1, i.e., k = 1, whence T(4,1) = 4. Three have k+1 = 3 <=> k = 2, so T(4,2) = 3. One has k+1 = 4, so T(4,3) = 1.
For n = 5, the solutions are, omitting initial components x_i = 1: {(6, 45), (7, 25), (9, 15), (10, 13), (2, 3, 35), (2, 5, 9), (3, 3, 10), (3, 5, 5)}. Therefore T(5,1..4) = (4, 4, 0, 0).
For n = 6, the solutions are (omitting x_i = 1): {(7, 66), (8, 36), (9, 26), (10, 21), (11, 18), (12, 16), (2, 4, 27), (2, 5, 15), (2, 6, 11), (2, 7, 9), (3, 3, 18), (3, 4, 10), (3, 6, 6), (2, 2, 2, 24), (2, 2, 3, 9), (2, 2, 4, 6), (2, 3, 3, 5)}. Therefore T(6,1..5) = (6, 7, 4, 0, 0).
For n = 9, the 27 solutions are (omitting '1's): {(10, 153), (11, 81), (12, 57), (13, 45), (15, 33), (17, 27), (18, 25), (21, 21), (2, 5, 117), (2, 6, 42), (2, 7, 27), (2, 9, 17), (2, 12, 12), (3, 4, 39), (3, 5, 21), (3, 6, 15), (3, 7, 12), (3, 9, 9), (4, 4, 18), (5, 5, 9), (6, 6, 6), (2, 2, 3, 36), (2, 2, 6, 9), (2, 3, 3, 13), (3, 3, 3, 7), (3, 3, 4, 5), (2, 3, 3, 3, 3)}. Therefore T(9,1..6) = (8, 13, 5, 1, 0, 0).

David A. Corneth, Table of n, a(n) for n = 2..14645 (first 899 rows flattened)
Peter Shiu, On Erdös's Last Equation, Amer. Math. Monthly, 126 (2019), 802-808.

Cf. A328910, A329202.

A328911(n,k,show=1)={if( k0 && (n*vecsum(x)+t)%d==0 && (n*vecsum(x)+t)\d >= x[k] && s++&& show&& printf("%d,",concat(x,(n*vecsum(x)+t)\d)),1);s)}

A329206 Minimal product of a solution to Erdős's Last Equation in n variables, x_1...x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

Original entry on oeis.org

16, 27, 48, 75, 90, 126, 128, 162, 240, 297, 288, 416, 378, 405, 432, 680, 540, 855, 640, 756, 880, 1242, 864, 1000, 1248, 1080, 1260, 1856, 1350, 2232, 1440, 1782, 2040, 1890, 1728, 2997, 2432, 2457, 2160, 3690, 2520, 4128, 2816, 2835, 3726, 4935

Offset: 2

Hugo Pfoertner, Nov 08 2019

nonn

Hugo Pfoertner, Table of n, a(n) for n = 2..121, extracted from David A. Corneth's list of solutions.
David A. Corneth, list of solutions and product of variables of solutions for n = 2..160, omitting ones.

Cf. A328910, A328911, A328980, A329205.

A329205 Lexicographically last solution to Erdős's Last Equation in n variables, x_1...x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n, written as triangle T(n,k), 1<=k<=n.

Original entry on oeis.org

1, 4, 4, 3, 3, 3, 2, 2, 2, 6, 1, 1, 3, 5, 5, 1, 1, 2, 3, 3, 5, 1, 1, 1, 2, 3, 3, 7, 1, 1, 2, 2, 2, 2, 2, 4, 1, 1, 1, 1, 2, 3, 3, 3, 3, 1, 1, 1, 1, 1, 2, 2, 2, 3, 10, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 11, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6

Offset: 1

Hugo Pfoertner, Nov 08 2019

a(1) = T(1,1) = 1 assumed. Any number is a solution for n=1.

			The triangle begins:
  1;
  4, 4;
  3, 3, 3;
  2, 2, 2, 6;
  1, 1, 3, 5, 5;
  1, 1, 2, 3, 3, 5;
  1, 1, 1, 2, 3, 3, 7;
  1, 1, 2, 2, 2, 2, 2, 4;

R. K. Guy, Sum equals product. in: Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, chapter D24, (2004), 299-301.

David A. Corneth, list of solutions and product of variables of solutions for n = 2..160, omitting ones.

Cf. A328910, A328980, A329206.

cp's OEIS Frontend

A328980 Erroneous version of A328910: Number of solutions to Erdős's Last Equation in n variables.

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A213820 Principal diagonal of the convolution array A213819.

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A328911 Irregular triangle read by rows: T(n,k) = number of solutions to Erdös's Last Equation x_1*...*x_n = n*(x_1+...+x_n), 0 < x_1 <= ... <= x_n, having k+1 components x_i > 1, 1 <= k <= 2*log_2(n).

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A329206 Minimal product of a solution to Erdős's Last Equation in n variables, x_1*...*x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

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A329205 Lexicographically last solution to Erdős's Last Equation in n variables, x_1*...*x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n, written as triangle T(n,k), 1<=k<=n.

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A328911 Irregular triangle read by rows: T(n,k) = number of solutions to Erdös's Last Equation x_1...x_n = n(x_1+...+x_n), 0 < x_1 <= ... <= x_n, having k+1 components x_i > 1, 1 <= k <= 2log_2(n).

A329206 Minimal product of a solution to Erdős's Last Equation in n variables, x_1...x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

A329205 Lexicographically last solution to Erdős's Last Equation in n variables, x_1...x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n, written as triangle T(n,k), 1<=k<=n.