A352620 -id:A352620 - cp's OEIS frontend

A003165 a(n) = floor(n/2) + 1 - d(n), where d(n) is the number of divisors of n.

0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 1, 5, 4, 4, 4, 7, 4, 8, 5, 7, 8, 10, 5, 10, 10, 10, 9, 13, 8, 14, 11, 13, 14, 14, 10, 17, 16, 16, 13, 19, 14, 20, 17, 17, 20, 22, 15, 22, 20, 22, 21, 25, 20, 24, 21, 25, 26, 28, 19, 29, 28, 26, 26, 29, 26, 32, 29, 31, 28, 34, 25, 35, 34, 32

Offset: 1

N. J. A. Sloane

a(n) is the number of partitions of n into exactly two parts whose smallest part is a nondivisor of n (see example). If n is prime, all of the smallest parts (except for 1) are nondivisors of n. Since there are floor(n/2) total partitions of n into two parts, then a(n) = floor(n/2) - 1 for primes (since we exclude 1). Proof: n = p implies a(p) = floor(p/2) + 1 - d(p) = floor(p/2) + 1 - 2 = floor(p/2) - 1. Furthermore, if n is an odd prime, a(n) = (n-3)/2. - Wesley Ivan Hurt, Jul 16 2014

a(n) is the nullity of the (n-1) X (n-1) matrix M(n) with entries M(n)[i,j] = i*j mod n (matrices given by A352620). - Luca Onnis, Mar 27 2022

			a(20) = 5. The partitions of 20 into exactly two parts are: (19,1), (18,2), (17,3), (16,4), (15,5), (14,6), (13,7), (12,8), (11,9), (10,10). Of these, there are exactly 5 partitions whose smallest part does not divide 20: {3,6,7,8,9}. - _Wesley Ivan Hurt_, Jul 16 2014

M. Newman, Integral Matrices. Academic Press, NY, 1972, p. 186.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A004526, A352620.

with(numtheory): A003165:=n->floor(n/2)+1-tau(n): seq(A003165(n), n=1..100); # Wesley Ivan Hurt, Jul 16 2014

Table[Floor[n/2]+1-DivisorSigma[0,n],{n,80}] (* Harvey P. Dale, May 09 2011 *)

a(n) = n\2 + 1 - numdiv(n); \\ Michel Marcus, Sep 18 2017

def A003165(n):
    return sum(1 for k in (1..n//2) if n % k)
[A003165(n) for n in (1..75)] # Peter Luschny, Jul 16 2014

a(n) = A004526(n) - A000005(n) + 1.
a(n) = Sum_{i=1..floor(n/2)} ceiling(n/i) - floor(n/i). - Wesley Ivan Hurt, Jul 16 2014
a(n) = Sum_{i=1..n} ceiling(n/i) mod floor(n/i). - Wesley Ivan Hurt, Sep 15 2017
G.f.: x*(1 + x - x^2)/((1 - x)^2*(1 + x)) - Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017
a(n) = Sum_{i=1..floor((n-1)/2)} sign((n-i) mod i). - Wesley Ivan Hurt, Nov 17 2017

More terms from Ralf Stephan, Sep 18 2004

A349099 a(n) is the permanent of the n X n matrix M(n) defined as M(n)[i,j] = i*j (mod n + 1).

Original entry on oeis.org

1, 1, 5, 32, 1074, 12600, 1525292, 34078720, 4072850100, 263459065600, 106809546673488, 2254519427530752, 3172225081523720416, 210351382651302645760, 45654014718074873700000, 11122845097194072534155264, 18156837198112938091803999360, 795289872611524024920215715840

Offset: 0

Stefano Spezia, Mar 25 2022

nonn

Det(M(n)) = 0 iff n = 4 or n > 5.
Rank(M(n)) = A088922(n+1).
Tr(M(n)) = A048153(n+1).

			See A352620 for the examples of matrix M(n).

Cf. A352620.

Cf. A048153, A074930, A088922, A160255.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
         Matrix(n, (i, j)-> (i*j) mod (n+1)))):
seq(a(n), n=0..16);  # Alois P. Heinz, Mar 25 2022

Join[{1},Table[Permanent[Table[Mod[j*Table[i, {i, n}], n+1], {j, n}]], {n, 17}]]

a(n) = matpermanent(matrix(n,n,i,j,(i*j)%(n+1))); \\ Michel Marcus, Mar 26 2022

A350710 Triangle read by rows formed from the coefficients in ascending order of the characteristic polynomial of the n X n matrix M(n) with entries M(n)[i,j] = i*j mod n+1.

Original entry on oeis.org

1, -1, 1, -3, -2, 1, -16, -16, -2, 1, 0, 100, -10, -10, 1, -1296, 0, 324, -24, -13, 1, 0, 0, 4116, 392, -175, -14, 1, 0, -131072, 16384, 12288, -512, -352, -12, 1, 0, 0, -708588, 0, 44469, 2592, -459, -24, 1, 0, 0, 16000000, 800000, -760000, -12000, 11000, -100, -45, 1

Offset: 0

Luca Onnis, Mar 27 2022

			Triangle begins:
n=0:     1;
n=1:    -1,   1;
n=2:    -3,  -2,    1;
n=3:   -16, -16,   -2,   1;
n=4:     0, 100,  -10, -10,    1;
n=5: -1296,   0,  324, -24,  -13,   1;
n=6:     0,   0, 4116, 392, -175, -14, 1;
For example, the characteristic polynomial associated to M(7) is
  q^7 - 12*q^6 - 352*q^5 - 512*q^4 + 12288*q^3 + 16384*q^2 - 131072*q + 0;
so the seventh row of the triangle is
  0, -131072, 16384, 12288, -512, -352, -12, 1.

Cf. A352620 (matrices M).

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(LinearAlgebra[
    CharacteristicPolynomial](Matrix(n, (i, j)-> irem(i*j, n+1)), x)):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 27 2022

Table[(-1)^(p + 1)*CoefficientList[CharacteristicPolynomial[Table[Mod[k*Table[i, {i, 1, p - 1}], p], {k, 1, p - 1}], x], x], {p, 2, 20}]

row(n) = Vecrev(charpoly(matrix(n,n,i,j,i*j%(n+1)))); \\ Kevin Ryde, Mar 27 2022

cp's OEIS Frontend

A003165 a(n) = floor(n/2) + 1 - d(n), where d(n) is the number of divisors of n.

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A349099 a(n) is the permanent of the n X n matrix M(n) defined as M(n)[i,j] = i*j (mod n + 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

A350710 Triangle read by rows formed from the coefficients in ascending order of the characteristic polynomial of the n X n matrix M(n) with entries M(n)[i,j] = i*j mod n+1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI