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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: T. Amdeberhan

T. Amdeberhan's wiki page.

T. Amdeberhan has authored 2 sequences.

A214043 Count of Laurent monomials (including multiplicities), in the Symplectic Schur symmetric polynomials s(mu, n) summed over all partitions mu of n.

Original entry on oeis.org

2, 15, 134, 1589, 20162, 293580, 4519916, 75850054, 1334978228, 24987138510, 487322528552, 9968005618302, 211338028257280, 4658444968474433, 105985325960653194, 2492041019432287042, 60271996071301852442, 1500054086883728030496
Offset: 1

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Author

T. Amdeberhan, Jul 13 2012

Keywords

Examples

			For n = 2, partition = (1, 1), the Symplectic Schur is: x_1*x_2 + x_1/x_2 + x_2/x_1 + 1/(x_1*x_2) + 1. There are five terms here. Partition (2) contributes another ten terms, including the term 1 twice. So a(2) = 5+10 = 15. [Extended by _Andrey Zabolotskiy_, Jan 24 2018]

Links

Programs

  • Mathematica
    s[mu_,n_] := Expand[Simplify[Det[Table[x[j]^(mu[[i]]+n-i+1) - x[j]^(-mu[[j]]-n+i-1), {i,n}, {j,n}]] / Det[Table[x[j]^(n-i+1) - x[j]^(-n+i-1), {i,n}, {j,n}]]]];
Table[Sum[s[PadRight[mu,n], n] /. {x[_]->1}, {mu, IntegerPartitions[n]}], {n, 5}]
(* Andrey Zabolotskiy, Jan 24 2018 *)

A080020 Primes of the form 9k^2 + 3k + 367, where k can be negative.

Original entry on oeis.org

367, 373, 379, 397, 409, 439, 457, 499, 523, 577, 607, 673, 709, 787, 829, 919, 967, 1069, 1123, 1237, 1297, 1423, 1489, 1627, 1699, 2089, 2347, 2437, 2719, 2917, 3019, 3229, 3559, 3673, 3907, 4027, 4273, 4657, 4789, 5059, 5197, 5479, 5623, 6067, 6373
Offset: 1

Views

Author

T. Amdeberhan, Jan 20 2003

Keywords

Comments

Original definition: Primes of the form q(n) = 370 + 18*binomial(ceiling(n/2), 2) + 3*(-1)^n*(2*ceiling(n/2) - 1).
The smallest positive k for which q(k) is not prime is k = 26.
Every q(k) is a divisor of some value of e(x) = x^2 + x + 41, the Euler prime-generating polynomial. Specifically, e(3*k^2 - 2*k + 122) = q(2*k) * e(k-1) and e(3*k^2 + 2*k + 122) = q(2*k + 1) * e(k).
Also primes of the form (k^2 + 1467)/4 with k odd. These primes are composite in O_Q(sqrt(-163)), since they can be expressed as (k/2 - 3*sqrt(-163))*(k/2 + 3*sqrt(-163)). For example, (7/2 - 3*sqrt(-163)/2)(7/2 + 3*sqrt(-163)/2) = 379. - Alonso del Arte, Nov 15 2017

Examples

			Given k = -2, we have 9 * 4 - 3 * 2 + 367 = 36 - 6 + 367 = 397 (a prime).
Given k = -1, we have 9 - 3 + 367 = 373 (a prime).
Given k = 0, we have 367 (a prime).
Given k = 1, we have 9 + 3 + 367 = 379 (a prime).
Given k = 2, we have 9 * 4 + 3 * 2 + 367 = 36 + 6 + 367 = 409 (a prime).

Links

Crossrefs

Cf. A005846.

Programs

  • Mathematica
    Union[Select[Table[9n^2 + 3n + 367, {n, -30, 30}], PrimeQ]] (* Harvey P. Dale, Mar 23 2013 *)

Formula

From Alonso del Arte, Nov 16 2017: (Start)
((6n - 1)^2 + 1467)/4 = (36n^2 - 12n + 1468)/2 = 9n^2 - 3n + 367.
((6n + 1)^2 + 1467)/4 = (36n^2 + 12n + 1468)/2 = 9n^2 + 3n + 367. (End)

Extensions

Edited by Dean Hickerson, Jan 20 2003
New definition from Charles R Greathouse IV, Feb 15 2011

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