A002324 -id:A002324 - cp's OEIS frontend

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A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

1, 1, 3, 3, 4, 3, 8, 5, 9, 4, 10, 9, 14, 8, 12, 11, 16, 9, 20, 12, 24, 10, 22, 15, 21, 14, 27, 24, 28, 12, 32, 21, 30, 16, 32, 27, 38, 20, 42, 20, 40, 24, 44, 30, 36, 22, 46, 33, 57, 21, 48, 42, 52, 27, 40, 40, 60, 28, 58, 36, 62, 32, 72, 43, 56, 30, 68, 48, 66, 32

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Claudia Rella, Resurgence, Stokes constants, and arithmetic functions in topological string theory, arXiv:2212.10606 [hep-th], 2022. See pages 21 - 23.

Cf. A000593, A002324, A050469, A078708, A326399, A326400.

Cf. A175646, A340577.

nmax = 70; CoefficientList[Series[Sum[k x^k/(1 + x^k + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 70}]
f[p_, e_] := Which[p == 3, p^e, Mod[p, 3] == 1, (p^(e + 1) - 1)/(p - 1), Mod[p, 3] == 2, (p^(e + 1) + (-1)^e)/(p + 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 3, 3^f[i,2], if(f[i,1]%3 == 1, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1), (f[i,1]^(f[i,2]+1) + (-1)^f[i,2])/(f[i,1] + 1))));} \\ Amiram Eldar, Nov 06 2022

a(n) = Sum_{d|n, n/d==1 (mod 3)} d - Sum_{d|n, n/d==2 (mod 3)} d.
a(n) = A326399(n) - A326400(n).
Multiplicative with a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1 (mod 3), and (p^(e+1) + (-1)^e)/(p + 1) if p == 2 (mod 3). - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) * Product_{primes p == 2 (mod 3)} 1/(1 + 1/p^2) = (1/2) * A175646 * (2*Pi^2/27)/A340577 = 0.3906512064... . - Amiram Eldar, Nov 06 2022

A364443 a(n) is the number of integers k of the form x^2 + x*y + y^2 (A003136) with n^2 < k < (n+1)^2.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 5, 4, 6, 5, 6, 8, 7, 8, 7, 9, 9, 11, 10, 10, 11, 10, 13, 12, 13, 13, 13, 14, 13, 16, 16, 16, 14, 16, 17, 16, 18, 20, 19, 19, 19, 19, 21, 20, 22, 21, 21, 22, 22, 24, 25, 21, 24, 25, 24, 27, 27, 25, 29, 26, 28, 26, 27, 29, 29, 30, 28, 29, 32, 31

Offset: 0

Hugo Pfoertner, Aug 05 2023

nonn

a(n) is the number of circles centered at (0,0) that pass through grid points of the hexagonal lattice that intersect the interior of an interval n < x < n+1 on the x-axis.

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
IBM Research, Circles on a triangular grid, Ponder This Challenge December 2023.
Hugo Pfoertner, Table of n, a(n) for n = 0..500000

Cf. A002324, A004016, A077773, A357112, A364729, A364730, A364731, A364732.

is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3);
for (k=0, 75, my (k1=k^2+1, k2=k^2+2*k, m=0); for (j=k1, k2, m+=is_a003136(j)); print1(m,", "))

from sympy import factorint
def A364443(n): return sum(1 for k in range(n**2+1,(n+1)**2) if not any(e&1 for p, e in factorint(k).items() if p % 3 == 2)) # Chai Wah Wu, Aug 07 2023

A035181 a(n) = Sum_{d|n} Kronecker(-9, d).

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 0, 4, 1, 4, 0, 3, 2, 0, 2, 5, 2, 2, 0, 6, 0, 0, 0, 4, 3, 4, 1, 0, 2, 4, 0, 6, 0, 4, 0, 3, 2, 0, 2, 8, 2, 0, 0, 0, 2, 0, 0, 5, 1, 6, 2, 6, 2, 2, 0, 0, 0, 4, 0, 6, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 3, 0, 0, 4, 0, 10, 1, 4, 0, 0, 4, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 6, 2, 2, 0, 9, 2, 4, 0, 8, 0

Offset: 1

N. J. A. Sloane

			x + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 4*x^8 + x^9 + 4*x^10 + 3*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A008441, A019693, A125079.

Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), A035184 (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -9, d], { d, Divisors[ n]}]] (* Michael Somos, Jun 24 2011 *)

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -9, d)))} \\ Michael Somos, Jun 24 2011

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -9, p) * X))) [n])} \\ Michael Somos, Jun 24 2011

{a(n) = local(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2, e+1, if( p==3, 1, if( p%4==1, e+1, (1 + (-1)^e)/2))))))} \\ Michael Somos, Jun 24 2011

A035181(n)=sumdivmult(n,d,kronecker(-9,d)) \\ M. F. Hasler, May 08 2018

From Michael Somos, Jun 24 2011: (Start)
a(n) is multiplicative with a(2^e) = e + 1, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4) and p > 3.
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker(-9, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker(-9, p) * p^-s)). (End)
a(3*n) = a(n). a(2*n + 1) = A125079(n). a(4*n + 1) = A008441(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.094395... (A019693). - Amiram Eldar, Oct 17 2022

A293899 Number of proper divisors of the form 3k+1 minus number of proper divisors of the form 3k+2.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 1, 1, -1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, -1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, 1, 2, -1, 1, 0, 1, 1, 0, -1, 1, 1, 2, 1, 0, 1, 1, 0, -1, 1, 2, -1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, -1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 2, -1, 1, 0, -1, 1, 0, 3, 1, 2, -1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0

Offset: 1

Antti Karttunen, Nov 06 2017

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002324, A010872, A073010, A293895, A293896.

Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 1, # != n] &] - DivisorSum[n, 1 &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)
Table[Total[Which[Mod[#,3]==1,1,Mod[#,3]==2,-1,True,0]&/@Most[ Divisors[ n]]],{n,110}] (* Harvey P. Dale, Nov 26 2021 *)

A293895(n) = sumdiv(n,d,(dA293896(n) = sumdiv(n,d,(dA293899(n) = (A293895(n) - A293896(n));

When n = 3k, a(n) = A002324(n), when n = 3k+1, a(n) = A002324(n) - 1, when n = 3k+2, a(n) = A002324(n) + 1.
a(n) = A002324(n) - A010872(n) (mod 3).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 25 2023

A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

Original entry on oeis.org

3, 7, 5, 11, 7, 13, 16, 9, 32, 17, 40, 11, 19, 55, 40, 24, 13, 23, 65, 69, 56, 25, 75, 15, 104, 32, 56, 29, 17, 87, 85, 119, 31, 72, 93, 64, 144, 19, 95, 133, 40, 136, 35, 105, 21, 105, 37, 111, 185, 88, 152, 176, 23, 80, 115, 161, 41, 123, 240, 48, 205, 240, 43, 25, 129, 175, 215, 88

Offset: 1

Bernard Schott, Sep 23 2022

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the a's.
For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions a of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, a is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with a = u^2 - v^2.
This sequence is not increasing. For example, a(2) = 7 for triangle with largest side = 13 while a(3) = 5 for triangle with largest side = 19.
Differs from A088514, the first 20 terms are the same then a(21) = 56 while A088514(21) = 25.
A229858 gives all the possible values of the smallest side a, in increasing order without repetition, but for all triples, not necessarily primitive.
All terms of A106505 are values taken by the smallest side a, in increasing order without repetition for primitive triples, but not all the lengths of this side a are present; example: 3 is not in A106505 (see comment in A229849).

			a(2) = a(5) = 7 because 2nd and 5th triple are respectively (7, 8, 13) and (7, 33, 37).

Cf. A357274 (triples), this sequence (smallest side), A357276 (middle side), A357277 (largest side), A357278 (perimeter).

Cf. also A002324, A050931, A088514, A106505, A229849.

for c from 5 to 181 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a, b)=1 and a

Formula

a(n) = A357274(n, 1).

A113063 Associated with theta series of hexagonal net with respect to a node.

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 2, 0, 2, 0, 0, 2, 2, 0, 0, 1, 0, 0, 2, 0, 4, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 2, 3, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 4, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0

Offset: 1

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Author

Michael Somos, Oct 13 2005

Keywords

nonn
easy
mult

Comments

Denoted by |lambda(n)| on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - Michael Somos, Jun 04 2015

Examples

G.f. = x + 2*x^3 + x^4 + 2*x^7 + 2*x^9 + 2*x^12 + 2*x^13 + x^16 + 2*x^19 + ...

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]

Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.

Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 5.

Crossrefs

Cf. A002324, A033687, A113062, A121839, A123477.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, 1, 1, -1, -1, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* Michael Somos, Jun 04 2015 *) f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[3, e_] := 2; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)

PARI

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, 1, 1, -1, -1, 1, -1][d%9 + 1]))};

PARI

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 2, p%6==1, e+1, !(e%2))))};

Formula

Moebius transform is period 9 sequence [ 1, -1, 1, 1, -1, -1, 1, -1, 0, ...].

a(n) is multiplicative with a(p^e) = 2 if p = 3 and e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 2, 5 (mod 6).

a(3*n + 2) = 0. a(3*n + 1) = A033687(n), a(3*n) = 2 * A002324(n).

3 * a(n) = A113062(n) unless n=0.

G.f.: Sum_{k>0} f(x^k) + f(x^(3*k)) where f(x) := x / (1 + x + x^2). - Michael Somos, Jun 04 2015

a(n) = |A123477(n)|. - Michael Somos, Dec 10 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(9*sqrt(3)) = 0.806133... (A121839 - 1). - Amiram Eldar, Dec 28 2023

A357112 a(n) = A035019(n)/6 for n > 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 4, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 4, 2, 1, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 4, 2, 2, 1, 2, 2, 2, 2, 1, 2, 4, 2, 1, 4, 2, 2, 4, 2, 2

Offset: 1

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Hugo Pfoertner, Sep 11 2022

Keywords

nonn

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002324, A003136, A004016, A343771.

A125096 Expansion of -1 + (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q.

Original entry on oeis.org

1, 0, 2, 2, 0, 0, 0, 2, 3, 0, 2, 4, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 6, 0, 0, 0, 0, 2, 0, 2, 4, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 6, 2, 0, 2, 4, 0, 0, 0, 0, 5, 0, 2, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 6, 2, 0, 0, 0, 0, 0

Offset: 1

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Author

Michael Somos, Nov 20 2006

Keywords

nonn
mult

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002324, A002325, A033761, A093954, A112603.

Programs

Mathematica

f[p_, e_] := If[MemberQ[{1, 3}, Mod[p, 8]], e + 1, (1 + (-1)^e)/2]; f[2, e_] := If[e > 1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 13 2022 *)

PARI

{a(n) = if( n<1, 0, qfrep([1, 0; 0, 8], n)[n] + qfrep([3, 1; 1, 3], n)[n])}

Formula

a(n) is multiplicative with a(2) = 0, a(2^e) = 2 if e>1, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8).

a(4*n + 2) = a(8*n + 5) = a(8*n + 7) = 0. a(4*n) = 2 * A002325(n). a(8*n + 1) = A112603(n). a(8*n + 3) = A033761(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(2)) = 1.110720... (A093954). - Amiram Eldar, Oct 13 2022

A157227 Number of primitive inequivalent (up to Pi/3 rotation) non-hexagonal sublattices of hexagonal (triangular) lattice of index n.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 2, 4, 4, 6, 4, 8, 4, 8, 8, 8, 6, 12, 6, 12, 10, 12, 8, 16, 10, 14, 12, 16, 10, 24, 10, 16, 16, 18, 16, 24, 12, 20, 18, 24, 14, 32, 14, 24, 24, 24, 16, 32, 18, 30, 24, 28, 18, 36, 24, 32, 26, 30, 20, 48, 20, 32, 32, 32, 28, 48, 22, 36, 32, 48

Offset: 1

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Author

N. J. A. Sloane, Feb 25 2009

Keywords

nonn

Links

John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 4.]

Crossrefs

Cf. A000086 (primitive hexagonal sublattices), A002324 (all hexagonal sublattices), A145394 (all sublattices), A001615, A304182.

Formula

a(n) = (A001615(n) - A000086(n))/3. - Andrey Zabolotskiy, May 09 2018

Extensions

New name and more terms from Andrey Zabolotskiy, May 09 2018

A344471 Number of points in the hexagonal lattice A_2 on the circle centered at the origin with squared radius A230655(n).

Original entry on oeis.org

1, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 384, 432, 576, 648, 768, 864, 960, 1152, 1296, 1536, 1728, 1920, 2304, 2592, 3072, 3456, 3840, 4608, 5184, 6144, 6912, 7680, 9216, 10368, 12288, 13824, 15360, 18432, 20736, 23040, 24576, 27648

Offset: 1

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Author

Jianing Song, May 20 2021

Keywords

nonn

Comments

Record values of A004016.

Examples

24 is a term because the circle with radius sqrt(91) centered at the origin hits exactly 24 points in the A_2 lattice, and any circle with radius < sqrt(91) centered at the origin hits fewer than 24 points.

Links

Jianing Song, Table of n, a(n) for n = 1..247

Crossrefs

Cf. A230655, A004016, A002324, A344472, A000005, A344473.

Cf. A071385 (similar sequence).

Programs

PARI

my(v=list_A344473(10^15), rec=0); print1(1, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(6*rec, ", "))) \\ see program for A344473

Formula

a(n) = A004016(A230655(n)).

a(n) = 6*A000005(A230655(n)) = 6*A002324(A230655(n)), n > 1.

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A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

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A364443 a(n) is the number of integers k of the form x^2 + x*y + y^2 (A003136) with n^2 < k < (n+1)^2.

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Python

A035181 a(n) = Sum_{d|n} Kronecker(-9, d).

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A293899 Number of proper divisors of the form 3k+1 minus number of proper divisors of the form 3k+2.

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Mathematica

PARI

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A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

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Maple

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A113063 Associated with theta series of hexagonal net with respect to a node.

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Examples

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Mathematica

PARI

PARI

Formula

A357112 a(n) = A035019(n)/6 for n > 0.

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A125096 Expansion of -1 + (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q.

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