A002324 -id:A002324 - cp's OEIS frontend

A096936 Half of number of integer solutions to the equation x^2 + 3y^2 = n.

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

Offset: 1

Michael Somos, Jul 15 2004

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

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.25).

Antti Karttunen, Table of n, a(n) for n = 1..65537
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
José Manuel Rodríguez Caballero, Divisors on overlapped intervals and multiplicative functions, arXiv:1709.09621 [math.NT], 2017.

Cf. A033716, A093766, A115979.

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then  a := a+1 ; end if; end do: a; end proc:
A002324 := proc(n) sigmamr(n,3,1)-sigmamr(n,3,2) ; end proc:
A096936 := proc(n) A002324(n) +2*(sigmamr(n,12,4)-sigmamr(n,12,8) ); end proc:
seq(A096936(n),n=1..90) ; # R. J. Mathar, Mar 23 2011

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1 || # == 3, 1, # == 2, 3 (1 + (-1)^#2)/2, Mod[#, 3] == 1, #2 + 1, True, (1 + (-1)^#2)/2] & @@@ FactorInteger[n])]; (* Michael Somos, Nov 20 2017 *)

{a(n) = if( n<1, 0, 1/2 * polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n)) * sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1 + x*O(x^n)), n))};

{a(n) = if( n<1, 0, qfrep([1, 0; 0, 3], n)[n])}; /* Michael Somos, Jun 05 2005 */

{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, 1, p==2, 3 * (1 + (-1)^e) / 2, p%3==2, (1 + (-1)^e) / 2, e+1)))}; /* Michael Somos, Nov 20 2017 */

(definec (A096936 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n)) (rest (A096936 (A028234 n)))) (cond ((= 2 p) (* (if (odd? e) 0 3) rest)) ((= 3 p) rest) ((= 1 (modulo p 3)) (* (+ 1 e) rest)) (else (* (if (odd? e) 0 1) rest)))))) ;; With the memoization-macro definec, after the given multiplicative formula. - Antti Karttunen, Nov 20 2017

a(n) = A033716(n) / 2.
Multiplicative with a(2^e) = 3*(1+(-1)^e)/2, a(3^e) = 1, a(p^e) = (1+(-1)^e)/2 if p==2 (mod 3) and a(p^e) = 1+e if p==1 (mod 3). - Corrected by Antti Karttunen, Nov 20 2017
G.f.: ((Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(3*k^2)) - 1)/2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Oct 15 2022

A145394 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/3 to give the other.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 4, 5, 5, 6, 4, 10, 6, 8, 8, 11, 6, 13, 8, 14, 12, 12, 8, 20, 11, 14, 14, 20, 10, 24, 12, 21, 16, 18, 16, 31, 14, 20, 20, 30, 14, 32, 16, 28, 26, 24, 16, 42, 21, 31, 24, 34, 18, 40, 24, 40, 28, 30, 20, 56, 22, 32, 36, 43, 28, 48, 24, 42, 32, 48, 24, 65, 26, 38, 42, 48, 32, 56, 28, 62

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

Also, apparently a(n) is the number of nonequivalent (up to lattice-preserving affine transformation) triangles on 2D square lattice of area n/2 [Karpenkov]. - Andrey Zabolotskiy, Jul 06 2017
From Andrey Zabolotskiy, Jan 18 2018: (Start)
The parent lattice of the sublattices under consideration has Patterson symmetry group p6, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A003051 (p6mm).
If we count sublattices related by parent-lattice-preserving reflection as equivalent, we get A003051 instead of this sequence. If we count sublattices related by rotation of the sublattice only (but not parent lattice; equivalently, sublattices related by rotation by angle which is not a multiple of Pi/3; see illustration in links) as equivalent, we get A054384. If we count sublattices related by any rotation or reflection as equivalent, we get A300651.
Rutherford says at p. 161 that a(n) != A054384(n) only when A002324(n) > 1, but actually these two sequences differ at other terms, too, for example, at n = 14 (see illustration). (End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Oleg Karpenkov, Elementary notions of lattice trigonometry, Mathematica Scandinavica, vol.102, no.2, pp.161-205, (2008) [See page 203].
Oleg Karpenkov, Geometry of Lattice Angles, Polygons, and Cones, Thesis, Technische Universität Graz, 2009.
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.
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 2.]
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A054384, A000203, A069734, A145391, A145392, A145393, A003051, A002324, A002654, A069735, A145390, A300651, A000086, A157227, A008611.

a[n_] := (DivisorSigma[1, n] + 2 DivisorSum[n, Switch[Mod[#, 3], 1, 1, 2, -1, 0, 0] &])/3; Array[a, 80] (* Jean-François Alcover, Dec 03 2015 *)

A002324(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)));
A000203(n) = if( n<1, 0, sigma(n));
a(n) = (A000203(n) + 2 * A002324(n)) / 3;
\\ Joerg Arndt, Oct 13 2013

a(n) = (A000203(n) + 2 * A002324(n))/3. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A000086(n/m^2) + A157227(n/m^2) = A002324(n) + Sum_{ m: m^2|n } A157227(n/m^2). [Rutherford] - Andrey Zabolotskiy, Apr 23 2018
a(n) = Sum_{ d|n } A008611(d-1). - Andrey Zabolotskiy, Aug 29 2019

New name from Andrey Zabolotskiy, Dec 14 2017

A343771 Smallest k such that circle centered at the origin and with radius sqrt(k) passes through exactly 6*n integer points in the hexagonal lattice (see A004016).

Original entry on oeis.org

1, 7, 49, 91, 2401, 637, 117649, 1729, 8281, 31213, 282475249, 12103, 13841287201, 1529437, 405769, 53599, 33232930569601, 157339, 1628413597910449, 593047, 19882681, 3672178237, 3909821048582988049, 375193, 68574961, 179936733613, 2989441, 29059303, 459986536544739960976801, 7709611

Offset: 1

Jianing Song, Apr 28 2021

nonn

a(n) is the smallest k such that A004016(k) = 6*n.
Also a(n) is the smallest index of n in A002324.
a(n) is the smallest term in A004611 that has exactly n divisors.

			91 = 7 * 13 is the smallest number all of whose prime factors are congruent to 1 modulo 3 with exactly 4 divisors, so a(4) = 91.
8281 = 7^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 3 with exactly 9 divisors, so a(9) = 8281.

Jianing Song, Table of n, a(n) for n = 1..1000
Helmut Ruhland, A family of lattices with an unbounded number of unit vectors, arXiv:2410.16172 [math.MG], 2024. See p. 2.

Cf. A004016, A002324, A004611, A230655, A355919.

Analog of A005179 and A018782.

primelist(d, r, l) = my(v=vector(l), i=0); if(l>0, forprime(p=2, oo, if(Mod(p, d)==r, i++; v[i]=p; if(i==l, break())))); v
prodR(n, maxf)=my(dfs=divisors(n), a=[], r); for(i=2, #dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)); for(j=1, #r, a=concat(a, [concat(dfs[i], r[j])]))))); a
A343771(n)=my(pf=prodR(n, n), a=1, b, v=primelist(3, 1, bigomega(n))); for(i=1, #pf, b=prod(j=1, length(pf[i]), v[j]^(pf[i][j]-1)); if(bA005179.

If the factorization of n into primes is n = Product_{i=1..r} p_i with p_1 >= p_2 >= ... >= p_r, then a(n) <= (q_1)^((p_1)-1) * (q_2)^((p_2)-1) * ... * (q_r)^((p_r)-1), where q_1 < q_2 < ... < q_r are the first r primes congruent to 1 modulo 3. The smallest n such that the equality does not hold is n = 128. [Those n such that the equality does not hold are listed in A355919. - Jianing Song, Jul 21 2022]
a(n) <= 7^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.
a(p*q) = 7^(p-1) * 13^(q-1) for primes p >= q.

A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002175, A008441, A019669, A053692, A113407, A121444.
Inverse Moebius transform of A034947.
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), this sequence (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_] := DivisorSum[n, KroneckerSymbol[-1, #] &]; Array[a, 105] (* Jean-François Alcover, Dec 02 2015 *)

{a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* Michael Somos, Jan 05 2012 */

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* Michael Somos, Jan 05 2012 */

a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - Michael Somos, Jan 05 2012
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).
Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - Ralf Stephan, Mar 27 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - Amiram Eldar, Oct 17 2022

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of triple [A010875(A020639(n)), A032742(n), A319710(n)] (with a separate value allotted for a(1)), or equally, of ordered pair [A319716(n), A319710(n)].
In addition to A319716, this filter sequence also records in the value of a(n) also the fact whether the smallest prime factor of n is unitary or not. This information is enough to determine the modulo 6 residues of all the divisors of n, thus sequences like A002324 are essentially functions of this sequence. Moreover, a lot of other information is immediately (and unavoidably) present, for example the exact prime signature of n, including also the relative order of exponents.
Any such filtering sequence can be perceived also in terms of what information it leaves out from a(n) that would be needed to reconstruct whole n from each a(n). If the whole n could be reconstructed from a(n) each time, then sequence a would be injective, and would be useless for filtering, because then it would match with any sequence. In this filter, what is left out is only the exact identity of the smallest prime factor, although its residue class mod 6 is retained. However, when the smallest prime factor is 2 or 3, this can be seen from that residue value, so for any number x in A047229, both A020639(x) and A032742(x) are known, and as x = A020639(x)*A032742(x), it means such numbers must occur in their own singleton equivalence classes.
Likewise, for any n in A283050, even if not divisible by 2 or 3, when we have A319710(n) stored in the triple as 1, this immediately gives away the exact identity of the smallest prime factor, which is equal to A014673(n) = A020639(A032742(n)) in these cases.
Thus there is a substantial subset of N (containing at least the union of A047229 and A283050) which is actually in the "blind sector" of this filter, "where anything goes", as this sequence obtains only unique values in that subdomain.
There is a related filter sequence A319996, which operates by "cleaving n from its high end" (by storing the residue class of the largest prime factor, A006530, instead of the smallest, together with n/A006530(n)), which has its own blind spots, but fortunately, they do not fully coincide with the blind spots of this filter. Naturally, any sequence like A002324 should match both to this sequence and A319996.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A319716(i) = A319716(j) => A319690(i) = A319690(j).

			For n = 65 = 5*13 and 143 = 11*13, the smallest prime factor is of the form 6k+5,  doesn't occur more than once in the factorization, and the largest proper divisor is the same number (13) in both cases, thus a(65) = a(143) (= 51, a running count value allotted by rgs-transform for this equivalence class).
For n = 1805 (5*19^2), 3971 (11*19^2), 6137 (17*19^2), it's like above, but the largest proper divisor is in all three cases 361 = 19^2, thus a(1805) = a(3971) = a(6137) (= 1405).
Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3, and also, they may not contain numbers from A283050. See the comments section.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A101296, A319707, A319716, A319996.
Cf. also A320004 (analogous sequence for modulo 4 residues).
Differs from A319707 for the first time at n=143, where a(143) = 51, differs from A319716 for the first time at n=121, where a(121) = 95.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286476(n) = if(1==n,n,(6*A032742(n) + (n % 6)));
A319710(n) = ((n>1)&&(factor(n)[1,2]>1));
v319717 = rgs_transform(vector(up_to,n,[A286476(n),A319710(n)]));
A319717(n) = v319717[n];

A320117 Filter sequence for counting the residue classes mod 6 of divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 10, 5, 18, 19, 12, 20, 21, 5, 22, 7, 23, 13, 10, 24, 25, 7, 12, 17, 26, 5, 27, 7, 16, 28, 10, 5, 29, 30, 31, 13, 21, 5, 32, 24, 33, 17, 10, 5, 34, 7, 12, 35, 36, 24, 22, 7, 16, 13, 37, 5, 38, 7, 12, 39, 21, 24, 27, 7, 40, 41, 10, 5, 42, 24, 12, 13, 26, 5, 43, 44, 16, 17, 10, 24, 45, 7, 46, 28, 47, 5, 22, 7, 33

Offset: 1

Antti Karttunen, Oct 06 2018

nonn

Restricted growth sequence transform of A320116.
For all i, j:
  A319717(i) = A319717(j) => a(i) = a(j),
  A319996(i) = A319996(j) => a(i) = a(j),
  A320113(i) = A320113(j) => a(i) = a(j),
  a(i) = a(j) => A002324(i) = A002324(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A319717, A319996, A320109, A320113, A320115, A320116.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A320116(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(1+(d%6)))); (m); };
v320117 = rgs_transform(vector(up_to,n,A320116(n)));
A320117(n) = v320117[n];

A123477 Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.

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

Michael Somos, Sep 27 2006

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Denoted by lambda(n) on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - Michael Somos, Dec 10 2017

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

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv preprint arXiv:1610.07793 [math.NT], 2016.

Cf. A002324, A005928, A033687, A113063.

A123477 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if modp(p,6) = 1 then
            a := a*(e+1) ;
        elif modp(p,6) in {2,5} then
            a := a*(1+(-1)^e)/2 ;
        elif e > 0 then
            a := -2*a ;
        end if;
    end do:
    a ;
end proc:
seq(A123477(n),n=1..100) ; # R. J. Mathar, Feb 22 2021

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, -3, 1, -1, 3, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* Michael Somos, Dec 10 2017 *)

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9+1]))};

{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))))};

Moebius transform is period 9 sequence [1, -1, -3, 1, -1, 3, 1, -1, 0, ...].
a(n) is multiplicative and 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) = A005928(n) unless n=0. |a(n)| = A113063(n).

A318982 a(n) = Sum_{d|n} Kronecker(-67, d).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0

Offset: 1

Jianing Song, Sep 06 2018

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -67.
Half of the number of integer solutions to x^2 + x*y + 17*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-67)] counted up to association.
Inverse Moebius transform of A011596.

			G.f. = x + x^4 + x^9 + x^16 + 2*x^17 + 2*x^19 + 2*x^23 + x^25 + 2*x^29 + x^36 + 2*x^37 + 2*x^47 + x^49 + 2*x^59 + x^64 + x^67 + 2*x^68 + 2*x^71 + 2*x^73 + 2*x^76 + ...

Jianing Song, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

Cf. A318984.
Moebius transform gives A011596.
Number of integral elements with norm n in Q[sqrt(d)] counted up to association: A002324 (d=-3), A002654 (d=-4), A035182 (d=-7), A002325 (d=-8), A035179 (d=-11), A035171 (d=-19), A035147 (d=-43), this sequence (d=-67), A318983 (d=-163).

a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-67, #] &]];
Table[a[n], {n, 1, 110}] (* Vincenzo Librandi, Sep 10 2018 *)

PARI
```
a(n) = sumdiv(n, d, kronecker(-67, d))
```

a(n) is multiplicative with a(67^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-67, p) = -1, a(p^e) = e + 1 if Kronecker(-67, p) = 1.
G.f.: Sum_{k>0} Kronecker(-67, k) * x^k / (1 - x^k).
A318984(n) = 2 * a(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(67) = 0.383806... . - Amiram Eldar, Dec 16 2023

A318983 a(n) = Sum_{d|n} Kronecker(-163, d).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0

Offset: 1

Jianing Song, Sep 06 2018

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -163.
Half of the number of integer solutions to x^2 + x*y + 41*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-163)] counted up to association.
Inverse Moebius transform of A011615.

			G.f. = x + x^4 + x^9 + x^16 + x^25 + x^36 + 2*x^41 + 2*x^43 + 2*x^47 + x^49 + 2*x^53 + 2*x^61 + x^64 + 2*x^71 + ...

Jianing Song, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

Cf. A318985.
Moebius transform gives A011615.
Number of integral elements with norm n in Q[sqrt(d)] counted up to association: A002324 (d=-3), A002654 (d=-4), A035182 (d=-7), A002325 (d=-8), A035179 (d=-11), A035171 (d=-19), A035147 (d=-43), A318982 (d=-67), this sequence (d=-163).

a[n_] := DivisorSum[n, KroneckerSymbol[-163, #] &]; Array[a, 100] (* Amiram Eldar, Dec 16 2023 *)

PARI
```
a(n) = sumdiv(n, d, kronecker(-163, d))
```

a(n) is multiplicative with a(163^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-163, p) = -1, a(p^e) = e + 1 if Kronecker(-163, p) = 1.
G.f.: Sum_{k>0} Kronecker(-163, k) * x^k / (1 - x^k).
A318985(n) = 2 * a(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(163) = 0.246068... . - Amiram Eldar, Dec 16 2023

A319996 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 6), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 10, 5, 18, 19, 12, 20, 21, 5, 22, 7, 23, 13, 10, 24, 25, 7, 12, 17, 26, 5, 27, 7, 16, 28, 10, 5, 29, 30, 31, 13, 21, 5, 32, 33, 34, 17, 10, 5, 35, 7, 12, 36, 37, 24, 22, 7, 16, 13, 38, 5, 39, 7, 12, 40, 21, 41, 27, 7, 42, 43, 10, 5, 44, 33, 12, 13, 26, 5, 45, 46, 16, 17, 10, 24, 47, 7, 48, 28, 49, 5, 22, 7, 34

Offset: 1

Antti Karttunen, Oct 05 2018

nonn

Restricted growth sequence transform of triple [A010875(A006530(n)), A052126(n), A319988(n)], with a separate value allotted for a(1).
Many of the same comments as given in A319717 apply also here, except for this filter, the "blind spot" area (where only unique values are possible for a(n)) is different, and contains at least all numbers in A070003. Because presence of 2 or 3 in the prime factorization of n do not force the value of a(n) unique, this is substantially less lax (i.e., more exact) filter than A319717. Here among the first 100000 terms, only 2393 have a unique value, compared to 74355 in A319717.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A319690(i) = A319690(j).

			For n = 15 (3*5) and n = 33 (3*11), the mod 6 residue of the largest prime factor is 5, also in both cases it is unitary (A319988(n) = 1), and the quotient n/A006530(n) is equal, in this case 3. Thus a(15) and a(33) are alloted the same running count (13 in this case) by rgs-transform.
For n = 2275 (5^2 * 7 * 13), n = 3325 (5^2 * 7 * 19), 5425 (5^2 * 7 * 31) and 6475 (5^2 * 7 * 37), the largest prime factor = 1 (mod 6), and A052126(n) = 175, thus these numbers are allotted the same running count (394 in this case) by rgs-transform.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A006530, A052126, A319988, A319717.
Cf. A007528 (positions of 5's), A002476 (of 7's), A112774 (after its initial term gives the position of 10's in this sequence).
Cf. also A319994 (modulo 4 analog).

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A319996aux(n) = if(1==n,0,[A006530(n)%6, A052126(n), A319988(n)]);
v319996 = rgs_transform(vector(up_to,n,A319996aux(n)));
A319996(n) = v319996[n];

cp's OEIS Frontend

A096936 Half of number of integer solutions to the equation x^2 + 3y^2 = n.

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