A006590 -id:A006590 - cp's OEIS frontend

A161886 Number of nonzero elements in the n X n Redheffer matrix.

1, 4, 7, 11, 14, 19, 22, 27, 31, 36, 39, 46, 49, 54, 59, 65, 68, 75, 78, 85, 90, 95, 98, 107, 111, 116, 121, 128, 131, 140, 143, 150, 155, 160, 165, 175, 178, 183, 188, 197, 200, 209, 212, 219, 226, 231, 234, 245, 249, 256, 261, 268, 271, 280, 285, 294, 299, 304

Offset: 1

Mats Granvik, Jun 21 2009

nonn

			The 4x4 Redheffer matrix:
  1,1,1,1
  1,1,0,0
  1,0,1,0
  1,1,0,1
contains 11 nonzero elements.

Cf. A143104, A006590, A000005.

A161886[n_] := Plus @@ Table[DivisorSigma[0, i], {i, 1, n}] + n - 1 (* Enrique Pérez Herrero, Sep 25 2009 *)
A161886[n_] := Total[Table[ Boole[Divisible[i, j] || (i == 1)], {i, 1, n}, {j, 1, n}], Infinity] (* Enrique Pérez Herrero, Sep 25 2009 *)
A161889[n_] := Plus @@ Plus @@ Table[Boole[Divisible[i, j] || (i == 1)], {i, 1, n}, {j, 1, n}] (* Enrique Pérez Herrero, Sep 28 2009 *)
A161889[n_] := Sum[Ceiling[n/i], {i, 1, n}] + DivisorSigma[0, n] - 1 (* Enrique Pérez Herrero, Sep 28 2009 *)

from math import isqrt
def A161886(n): return (lambda m: 2*sum(n//k for k in range(1, m+1))+n-1-m*m)(isqrt(n)) # Chai Wah Wu, Oct 09 2021

a(n) = A006590(n)+A000005(n)-1. [Enrique Pérez Herrero, Sep 28 2009]
a(n) = A006218(n)+n-1. [Enrique Pérez Herrero, Sep 25 2009]
a(1) = 1, a(n) = a(n-1) + A000005(n) + 1 for n > 1. a(1) = 1, a(n) = A006218(n+1) - A000005(n+1) + n - 1 = A006218(n+1) + A049820(n+1) - 2 = A006590(n+1) - 2 for n > 1. [Jaroslav Krizek, Nov 08 2009]

Edited by N. J. A. Sloane, Jun 26 2009

A092494 a(n) = Sum_{p prime and p<=n} ceiling(n/p).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 10, 11, 12, 13, 16, 17, 20, 21, 23, 25, 27, 28, 31, 32, 34, 36, 39, 40, 42, 43, 45, 46, 49, 50, 54, 55, 56, 58, 60, 62, 65, 66, 68, 70, 73, 74, 78, 79, 81, 83, 86, 87, 89, 90, 92, 94, 97, 98, 100, 102, 104, 106, 109, 110, 114, 115, 117, 119, 120, 122

Offset: 1

Reinhard Zumkeller, Apr 05 2004

nonn

a(n) = A013939(n) + A048865(n).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006590.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
p:= 0:
do
  p:= nextprime(p);
  if p > N then break fi;
  V[p]:= V[p]+1;
  for k from 2 to floor(N/p) do
    V[(k-1)*p+1 .. k*p]:= V[(k-1)*p+1 .. k*p] +~ k;
  od;
  if (k-1)*p+1<=N then V[(k-1)*p+1..N]:= V[(k-1)*p+1..N]+~ k fi
od:
convert(V,list); # Robert Israel, Jun 19 2019

a(n) = sum(k=1, n, isprime(k)*ceil(n/k)); \\ Michel Marcus, Jun 19 2019

A309097 Number of partitions of n avoiding the partition (4,2,1).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 32, 39, 49, 56, 68, 79, 91, 103, 119, 132, 150, 165, 183, 202, 224, 241, 264, 287, 311, 334, 362, 385, 415, 442, 472, 503, 535, 563, 599, 634, 670, 703, 743, 778, 820, 859, 899, 942, 988, 1027, 1074, 1119, 1167, 1214, 1266

Offset: 0

Jonathan S. Bloom, Jul 12 2019

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.

Conjecture: for n > 0, a(n) is the number of ordered pairs (r, l) such that there exists a nilpotent matrix of order n whose rank is r and nilpotent index is l. Actually, such a matrix exists if and only if ceiling(n/(n-r)) <= l <= r+1, see my proof below. If this conjecture is true, then a(n) = (n^2 + 3n)/2 - A006590(n) for n > 0. - Jianing Song, Nov 04 2019

Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
J. Bloom and D. Saracino, On Criteria for rook equivalence of Ferrers boards, arXiv:1808.04221 [math.CO], 2018.
J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, arXiv:1808.04238 [math.CO], 2018.
J. Bloom and D. Saracino Rook and Wilf equivalence of integer partitions, European J. Combin., 76 (2018), 199-207.
J. Bloom and D. Saracino On Criteria for rook equivalence of Ferrers boards, European J. Combin., 71 (2018), 246-267.
Jianing Song, Proof that there exists a nilpotent matrix of order n whose rank is r and nilpotent index is l if and only if ceiling(n/(n-r)) <= l <= r+1
Wikipedia, Nilpotent matrix

Cf. A309098, A309099, A309058.

lista(n)=my(b(k)=x^k/(1-x^k)+O(x*x^n));Vec(1+sum(i=1,n,b(i)*(1+sum(j=i+1,n,b(j)*(1+b(j+1)))))) \\ Christian Sievers, Sep 01 2025

G.f.: 1 + Sum_{i>=1} b(i) * ( 1 + Sum_{j>i} b(j) * ( 1 + b(j+1) ) ) where b(k)=x^k/(1-x^k). - Christian Sievers, Sep 01 2025

More terms from Alois P. Heinz, Jul 12 2019

A332682 a(n) = Sum_{k=1..n} (-1)^(k+1) * ceiling(n/k).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 6, 5, 7, 8, 9, 8, 9, 10, 13, 11, 12, 13, 14, 13, 16, 17, 18, 15, 17, 18, 21, 20, 21, 22, 23, 20, 23, 24, 27, 25, 26, 27, 30, 27, 28, 29, 30, 29, 34, 35, 36, 31, 33, 34, 37, 36, 37, 38, 41, 38, 41, 42, 43, 40, 41, 42, 47, 43, 46, 47, 48, 47, 50

Offset: 1

Ilya Gutkovskiy, Feb 19 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006590, A048272, A059851, A275495, A325937.

f:= proc(n) local k; add((-1)^(k+1)*ceil(n/k),k=1..n) end proc:
map(f, [$1..100]); # Robert Israel, Nov 25 2024

Table[Sum[(-1)^(k + 1) Ceiling[n/k], {k, 1, n}], {n, 1, 70}]
nmax = 70; CoefficientList[Series[(x/(1 - x)) (1 + Sum[x^k/(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, (-1)^(k+1)*ceil(n/k)); \\ Michel Marcus, Feb 21 2020

G.f.: (x/(1 - x)) * (1 + Sum_{k>=2} x^k / (1 + x^k)).
G.f.: (x/(1 - x)) * (1 + Sum_{k>=1} (-1)^(k+1) * x^(2*k) / (1 - x^k)).
a(n) = (n mod 2) + Sum_{k=1..n-1} A048272(k).
a(n) = 1 + Sum_{k<=n-1} A325937(k). - Robert Israel, Nov 25 2024

A332623 a(n) = Sum_{k=1..n} ceiling(n/k)^2.

Original entry on oeis.org

1, 5, 14, 25, 43, 58, 87, 106, 141, 171, 212, 239, 302, 333, 386, 439, 507, 546, 631, 674, 765, 834, 911, 962, 1091, 1157, 1246, 1331, 1450, 1513, 1666, 1733, 1866, 1967, 2080, 2181, 2373, 2452, 2577, 2694, 2883, 2970, 3171, 3262, 3437, 3600, 3749, 3848, 4107, 4225

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

nonn

Cf. A000005, A000203, A006218, A006590, A024916, A222548, A332490, A332569, A332624.

[&+[Ceiling(n/k)^2:k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 17 2020

Table[Sum[Ceiling[n/k]^2, {k, 1, n}], {n, 1, 50}]
Table[n + Sum[2 DivisorSigma[1, k] + DivisorSigma[0, k], {k, 1, n - 1}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[x/(1 - x)^2 + x/(1 - x) Sum[(2 k + 1) x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

from math import isqrt
def A332623(n): return n-(s:=isqrt(n-1))**2*(s+2)+sum((q:=(n-1)//k)*((k<<1)+q+3) for k in range(1,s+1)) # Chai Wah Wu, Oct 24 2023

G.f.: x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} (2*k + 1) * x^k / (1 - x^k).
a(n) = n + Sum_{k=1..n-1} (2*sigma(k) + d(k)).
a(n) ~ n^2 * Pi^2 / 6. - Vaclav Kotesovec, Feb 20 2020

A332569 a(n) = Sum_{k=1..n} floor(n/k) * ceiling(n/k).

Original entry on oeis.org

1, 5, 12, 23, 36, 54, 74, 97, 125, 156, 186, 226, 268, 306, 354, 409, 458, 515, 574, 636, 710, 778, 838, 922, 1013, 1086, 1168, 1264, 1350, 1452, 1556, 1651, 1762, 1864, 1966, 2105, 2234, 2332, 2448, 2594, 2726, 2864, 3004, 3132, 3294, 3444, 3564, 3736, 3917, 4067

Offset: 1

Ilya Gutkovskiy, Feb 16 2020

nonn

Row sums of A085383.

Cf. A000203, A006218, A006590, A024916, A049697, A222548, A332490.

[&+[Floor(n/k)*Ceiling(n/k):k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 16 2020

Table[Sum[Ceiling[n/k] Floor[n/k], {k, 1, n}], {n, 1, 50}]
Table[1 + Sum[DivisorSigma[1, k] + DivisorSigma[1, k + 1], {k, 1, n - 1}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[((1 + x)/(1 - x)) Sum[x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, my(q=n/k); floor(q) * ceil(q)); \\ Michel Marcus, Feb 17 2020

from math import isqrt
from sympy import divisor_sigma
def A332569(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-divisor_sigma(n) # Chai Wah Wu, Oct 23 2023

G.f.: ((1 + x) / (1 - x)) * Sum_{k>=1} x^k / (1 - x^k)^2.
a(n) = 1 + Sum_{k=1..n-1} (sigma(k) + sigma(k+1)) for n > 0.
a(n) ~ (Pi*n)^2/6. - Vaclav Kotesovec, Jun 24 2021

A333465 a(n) = Sum_{k=1..n} ceiling(n/k) * gcd(n,k).

Original entry on oeis.org

1, 4, 8, 14, 17, 30, 27, 43, 47, 62, 48, 97, 60, 96, 114, 123, 83, 167, 95, 195, 177, 170, 119, 283, 181, 209, 230, 300, 158, 401, 172, 330, 308, 288, 348, 517, 213, 329, 377, 560, 239, 613, 253, 522, 599, 413, 279, 776, 415, 624, 520, 640, 322, 793, 604, 854, 594, 543, 364, 1220

Offset: 1

Ilya Gutkovskiy, Mar 22 2020

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A000005, A000010, A006590, A018804, A333463.

f:= n -> add(ceil(n/k)*igcd(n,k),k=1..n):
map(f, [$1..100]); # Robert Israel, Mar 24 2020

Table[Sum[Ceiling[n/k] GCD[n, k], {k, n}], {n, 60}]
Table[Sum[EulerPhi[n/d] (d + Sum[DivisorSigma[0, k], {k, d - 1}]), {d, Divisors[n]}], {n, 60}]

a(n) = sum(k=1, n, ceil(n/k)*gcd(n, k)); \\ Michel Marcus, Mar 23 2020

a(n) = Sum_{d|n} phi(n/d) * A006590(d).

a(n) = A018804(n) + Sum_{k=1..n-1} Sum_{d|k} gcd(n,d).

A332624 a(n) = Sum_{k=1..n} ceiling(n/k)^n.

Original entry on oeis.org

1, 5, 36, 289, 3433, 47578, 842499, 16850338, 389415029, 10010878371, 285679026506, 8918295095267, 302973286652448, 11112691430262573, 437929106387544254, 18447028378472722051, 827256956775203666857, 39346558275376372606086, 1978429667078835508142129

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

nonn

Cf. A006590, A031971, A332469, A332490, A332623.

[&+[Ceiling(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020

Table[Sum[Ceiling[n/k]^n, {k, 1, n}], {n, 1, 19}]
Table[n + Sum[Sum[(d + 1)^n - d^n, {d, Divisors[k]}], {k, 1, n - 1}], {n, 1, 19}]
Table[SeriesCoefficient[x/(1 - x)^2 + x/(1 - x) Sum[((k + 1)^n - k^n) x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 19}]

a(n) = [x^n] x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} ((k + 1)^n - k^n) * x^k / (1 - x^k).

a(n) = n + Sum_{k=1..n-1} Sum_{d|k} ((d + 1)^n - d^n).

A332846 a(1) = 1; a(n+1) = Sum_{k=1..n} a(k) * ceiling(n/k).

Original entry on oeis.org

1, 1, 3, 8, 20, 50, 121, 297, 716, 1739, 4198, 10157, 24513, 59246, 143006, 345381, 833792, 2013272, 4860337, 11734717, 28329772, 68396030, 165121957, 398644144, 962410246, 2323475153, 5609360573, 13542220814, 32693802921, 78929886033, 190553574988, 460037180829, 1110627936647

Offset: 1

Ilya Gutkovskiy, Feb 26 2020

nonn

Cf. A006590, A014668, A097919, A332490.

a[1] = 1; a[n_] := a[n] = Sum[a[k] Ceiling[(n - 1)/k], {k, 1, n - 1}]; Table[a[n], {n, 1, 33}]
a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] + Sum[a[d], {d, Divisors[k]}], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
terms = 33; A[] = 0; Do[A[x] = x (1 + (1/(1 - x)) (A[x] + x Sum[A[x^k], {k, 1, terms}])) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * (A(x) + x * Sum_{k>=1} A(x^k))).
a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-2} (a(k) + Sum_{d|k} a(d)).
a(n) ~ c * (1 + sqrt(2))^n, where c = 0.2594006517235012546870541901936538347053403598092060748627156661727... - Vaclav Kotesovec, Mar 10 2020

A333505 a(n) = Sum_{k=1..n} (-1)^(k+1) * k * ceiling(n/k).

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 5, 5, 1, 4, 9, 9, 2, 2, 9, 17, 5, 5, 11, 11, 2, 12, 23, 23, -4, 1, 14, 26, 15, 15, 22, 22, -6, 8, 25, 37, 9, 9, 28, 44, 7, 7, 18, 18, 3, 35, 58, 58, -9, -2, 18, 38, 21, 21, 36, 52, 5, 27, 56, 56, -3, -3, 28, 68, 8, 26, 45, 45, 24, 50, 73, 73, -23, -23, 14

Offset: 1

Ilya Gutkovskiy, May 25 2020

sign

Cf. A001057, A002129, A006590, A024916, A024919, A332490, A332682.

Table[Sum[(-1)^(k + 1) k Ceiling[n/k], {k, 1, n}], {n, 1, 75}]
Table[(-1)^(n + 1) Ceiling[n/2] + Sum[DivisorSum[k, (-1)^(# + 1) # &], {k, 1, n - 1}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[x/(1 - x) (1/(1 + x)^2 + Sum[(-1)^(k + 1) k x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, (-1)^(k+1)*k*ceil(n/k)); \\ Michel Marcus, May 26 2020

from math import isqrt
def A333505(n): return ((s:=isqrt(m:=n-1>>1))**2*(s+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1))<<1)-((t:=isqrt(n-1))**2*(t+1)-sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,t+1))>>1) + (m+1 if n&1 else -m-1) # Chai Wah Wu, Oct 30 2023

G.f.: (x/(1 - x)) * (1/(1 + x)^2 + Sum_{k>=1} (-1)^(k+1) * k * x^k / (1 - x^k)).
a(n) = (-1)^(n+1) * ceiling(n/2) + Sum_{k=1..n-1} A002129(k).
a(n) = A001057(n) - A024919(n-1).

cp's OEIS Frontend

A161886 Number of nonzero elements in the n X n Redheffer matrix.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A092494 a(n) = Sum_{p prime and p<=n} ceiling(n/p).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

A309097 Number of partitions of n avoiding the partition (4,2,1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A332682 a(n) = Sum_{k=1..n} (-1)^(k+1) * ceiling(n/k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A332623 a(n) = Sum_{k=1..n} ceiling(n/k)^2.

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A332569 a(n) = Sum_{k=1..n} floor(n/k) * ceiling(n/k).

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A333465 a(n) = Sum_{k=1..n} ceiling(n/k) * gcd(n,k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A332624 a(n) = Sum_{k=1..n} ceiling(n/k)^n.

Views

Author

Keywords