A255499 -id:A255499 - cp's OEIS frontend

A229738 a(n) = p^2(p^2+2p-1)/2, where p = prime(n).

14, 63, 425, 1519, 8591, 16393, 46529, 71839, 151823, 377609, 491071, 987049, 1480961, 1787983, 2542559, 4092713, 6262319, 7148041, 10374079, 13061231, 14585473, 19964959, 24297503, 32072129, 45172609, 53055401, 57362863, 66759119, 71868169, 82960193, 132112639, 149489471, 178699649, 189326479, 249739049

Offset: 1

N. J. A. Sloane, Oct 05 2013

nonn

Bruno Berselli, Table of n, a(n) for n = 1..1000
L. Kaylor, D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
Michael Knapp, Two by Two Matrices with Both Eigenvalues in Z/pZ, Math. Mag., Vol. 79, No. 2, April 2006.
G. Olsavsky, The Number of 2 by 2 Matrices over Z/pZ with Eigenvalues in the Same Field, Math. Mag., 76 (2003), 314-317.

Cf. A229739, A255499.

[p^2*(p^2+2*p-1)/2: p in PrimesUpTo(200)]; // Bruno Berselli, Oct 07 2013

Table[Prime[n]^2 (Prime[n]^2 + 2 Prime[n] - 1)/2, {n, 40}] (* Bruno Berselli, Oct 07 2013 *)
#^2 (#^2+2#-1)/2&/@Prime[Range[40]] (* Harvey P. Dale, Mar 13 2017 *)

a(n)=p=prime(n);p^2*(p^2+2*p-1)/2 \\ Anders Hellström, Sep 04 2015

lista(nn) = forprime(p=2, nn, print1(p^2*(p^2+2*p-1)/2, ", ")); \\ Michel Marcus, Sep 04 2015

A345018 For each n, append to the sequence n^2 consecutive integers, starting from n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41

Offset: 1

Paolo Xausa, Jun 05 2021

Irregular triangle read by rows T(n,k) in which row n lists the integers from n to n + n^2 - 1, with n >= 1.

			Written as an irregular triangle T(n,k) the sequence begins:
  n\k|  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 ...
  ---+---------------------------------------------------------------
   1 |  1;
   2 |  2,  3,  4,  5;
   3 |  3,  4,  5,  6,  7,  8,  9, 10, 11;
   4 |  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10416 (rows 1..31 of the triangle, flattened)

Column 1: A000027.
Right border: A028387.
Row lengths: A000290.
Row sums: A255499.
Cf. A064866, A074279.

T:= n-> (t-> seq(n+i, i=0..t-1))(n^2):
seq(T(n), n=1..6);  # Alois P. Heinz, Nov 05 2024

Table[Range[n,n^2+n-1],{n,6}] (* Paolo Xausa, Sep 05 2023 *)

row(n) = vector(n^2, k, n+k-1); \\ Michel Marcus, Jun 08 2021

from sympy import integer_nthroot
def A345018(n): return n-1+(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k*(3-(k<<1))+5)//6 # Chai Wah Wu, Nov 05 2024

T(n,k) = n + k - 1, with n >= 1 and 1 <= k <= n^2.

cp's OEIS Frontend

A229738 a(n) = p^2(p^2+2p-1)/2, where p = prime(n).

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A345018 For each n, append to the sequence n^2 consecutive integers, starting from n.

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cp's OEIS Frontend

A229738 a(n) = p^2*(p^2+2*p-1)/2, where p = prime(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

A345018 For each n, append to the sequence n^2 consecutive integers, starting from n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A229738 a(n) = p^2(p^2+2p-1)/2, where p = prime(n).