A343771 -id:A343771 - cp's OEIS frontend

A355919 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order; let b(n) = 7^(p_1 - 1)13^(p_2 - 1)19^(p_3 - 1)...*A002476(k)^(p_k - 1). Sequence lists m such that b(m) > A343771(m).

128, 256, 512, 1024, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 73728, 98304, 131072, 147456, 196608, 262144, 294912

Offset: 1

Jianing Song, Jul 20 2022

{b(n)} is an analog of A037019 and of A340388: all prime factors of b(n) are all congruent to 1 modulo 6 and b(n) has exactly n divisors, so A002324(b(n)) = n. By definition we have A343771(n) <= b(n), and it seems that the equality holds for most n. This sequence lists the exceptions.
Since {b(n)} agrees with A343771(n) for most n, it cannot have its own entry.
Let q be a prime, then q^e is here if and only if e >= N+1, where N is the number of primes congruent to 1 modulo 6 below 7^q (N = 6, 32, 958, ... for q = 2, 3, 5, ...).
Proof: p_1 < p_2 < ... be the primes congruent to 1 modulo 6. Suppose that A343771(q^e) = (p_1)^(q^(m_1)-1) * (p_2)^(q^(m_2)-1) * ... * (p_r)^(q^(m_r)-1) with r <= e, m_1 >= m_2 >= ... >= m_r. If m_1 >= 2, then r < e, so we can substitute (p_1)^(q^(m_1)-1) with (p_1)^(q^(m_1-1)-1) * (p_{r+1})^(q-1), which a smaller number with exactly q^e divisors, a contradiction. So we have m_1 = 1, namely A343771(q^e) = b(q^e). On the other hand, if e >= N+1, then A343771(q^e) <= (p_1)^(q^2-1) * (p_2)^(q-1) * ... * (p_{e-1})^(q-1) < b(q^e).
It seems that q^(N+1) is the smallest q-rough term in this sequence.

			128 is a term since b(128) = 7 * 13 * 19 * 31 * 37 * 43 * 61 > A343771(128) = 7^3 * 13 * 19 * 31 * 37 * 43.

Analog of A072066 and A340624.

Cf. A343771, A002476, A037019, A340388, A002324.

b(n) = my(f=factor(n), w=omega(n), p=1, product=1); forstep(i=w, 1, -1, for(j=1, f[i, 2], p=nextprime(p+1); while(!(p%6==1), p=nextprime(p+1)); product *= p^(f[i, 1]-1))); product
isA355919(n) = (b(n) > A343771(n)) \\ See A343771 for its program

a(20)-a(22) from Jinyuan Wang, Aug 10 2022

A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -3. See Formula section for the general expression. - N. J. A. Sloane, Mar 22 2022
Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)*p^(-s) + Kronecker(m,p) * p^(-2s))^(-1) for m = -3.
(Number of points of norm n in hexagonal lattice) / 6, n>0.
The hexagonal lattice is the familiar 2-dimensional lattice (A_2) in which each point has 6 neighbors. This is sometimes called the triangular lattice.
The first occurrence of a(n) = 1, 2, 3, 4,... is at n= 1, 7, 49, 91, 2401, 637, ... as tabulated in A343771. - R. J. Mathar, Sep 21 2024

			G.f. = x + x^3 + x^4 + 2*x^7 + x^9 + x^12 + 2*x^13 + x^16 + 2*x^19 + 2*x^21 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 112, first display.
J. W. L. Glaisher, Table of the excess of the number of (3k+1)-divisors of a number over the number of (3k+2)-divisors, Messenger Math., 31 (1901), 64-72.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
Hershel M. Farkas, On an arithmetical function, Ramanujan J., 8(3) (2004), 309-315.
Pavel Guerzhoy and Ka Lun Wong, Farkas' identities with quartic characters, arXiv:1905.06506 [math.NT], 2019.
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. [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 below.]
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.
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
José Manuel Rodríguez Caballero, Divisors on overlapped intervals and multiplicative functions, arXiv:1709.09621 [math.NT], 2017.
J. S. Rutherford, Generating functions for the cage isomers of the C_{20n} icosahedral fullerenes, J. Mathematical Chem., 14 (1993), 385-390. [From _N. J. A. Sloane_, Mar 12 2009]
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 1]. - From _N. J. A. Sloane_, Feb 23 2009
N. J. A. Sloane, Maple code for Dedekind zeta functions of quadratic number fields.

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A004016, A035019, A073010, A145377, A293899, A000086, A003136, A034020, A145394.

a002324 n = a001817 n - a001822 n  -- Reinhard Zumkeller, Nov 26 2011

A002324 := 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 p = 3 then
            ;
        elif modp(p,3) = 1 then
            a := a*(e+1) ;
        else
            a := a*(1+(-1)^e)/2 ;
        end if;
    end do:
    a ;
end proc:
seq(A002324(n),n=1..100) ; # R. J. Mathar, Sep 21 2024

dn12[n_]:=Module[{dn=Divisors[n]},Count[dn,?(Mod[#,3]==1&)]-Count[ dn,?(Mod[#,3]==2&)]]; dn12/@Range[120]  (* Harvey P. Dale, Apr 26 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Aug 24 2014 *)
Table[DirichletConvolve[DirichletCharacter[3,2,m],1,m,n],{n,1,30}] (* Steven Foster Clark, May 29 2019 *)
f[3, p_] := 1; f[p_, e_] := If[Mod[p, 3] == 1, e+1, (1+(-1)^e)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

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

{a(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)))};

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==3, 1, if( p%3==1, e+1, !(e%2))))))}; \\ Michael Somos, May 20 2005

{a(n) = if( n<1, 0, qfrep([2,1; 1,2], n, 1)[n] / 3)}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker(-3, p)*X))[n])}; \\ Michael Somos, Jun 05 2005

my(B=bnfinit(x^2+x+1)); vector(100,n,#bnfisintnorm(B,n)) \\ Joerg Arndt, Jun 01 2024

from math import prod
from sympy import factorint
def A002324(n): return prod(e+1 if p%3==1 else int(not e&1) for p, e in factorint(n).items() if p != 3) # Chai Wah Wu, Nov 17 2022

From N. J. A. Sloane, Mar 22 2022 (Start):
The Dedekind zeta function DZ_K(s) for a quadratic field K of discriminant D is as follows.
Here m is defined by K = Q(sqrt(m)) (so m=D/4 if D is a multiple of 4, otherwise m=D).
DZ_K(s) is the product of three terms:
(a) Product_{odd primes p | D} 1/(1-1/p^s)
(b) Product_{odd primes p such that (D|p) = -1} 1/(1-1/p^(2s))
(c) Product_{odd primes p such that (D|p) = 1} 1/(1-1/p^s)^2
and if m is
   0,1,2,3,4,5,6,7 mod 8, the prime 2 is to be included in term
   -,c,a,a,-,b,a,a, respectively.
For Maple (and PARI) implementations, see link. (End)
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 3*v^2 + 4*w^2 - 2*u*w + w - v. - Michael Somos, Jul 20 2004
Has a nice Dirichlet series expansion, see PARI line.
G.f.: Sum_{k>0} x^k/(1+x^k+x^(2*k)). - Vladeta Jovovic, Dec 16 2002
a(3*n + 2) = 0, a(3*n) = a(n), a(3*n + 1) = A033687(n). - Michael Somos, Apr 04 2003
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 - u3)*(u3 - u6) - (u2 - u6)^2. - Michael Somos, May 20 2005
Multiplicative with a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3). - Michael Somos, May 20 2005
G.f.: Sum_{k>0} x^(3*k - 2) / (1 - x^(3*k - 2)) - x^(3*k - 1)  / (1 - x^(3*k - 1)). - Michael Somos, Nov 02 2005
G.f.: Sum_{n >= 1} q^(n^2)(1-q)(1-q^2)...(1-q^(n-1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). - Jeremy Lovejoy, Jun 12 2009
a(n) = A001817(n) - A001822(n). - R. J. Mathar, Mar 31 2011
A004016(n) = 6*a(n) unless n=0.
Dirichlet g.f.: zeta(s)*L(chi_2(3),s), with chi_2(3) the nontrivial Dirichlet character modulo 3 (A102283). - Ralf Stephan, Mar 27 2015
From Andrey Zabolotskiy, May 07 2018: (Start)
a(n) = Sum_{ m: m^2|n } A000086(n/m^2).
a(A003136(m)) > 0, a(A034020(m)) = 0 for all m. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Oct 11 2022

More terms from David Radcliffe

Somos D.g.f. replaced with correct version by Ralf Stephan, Mar 27 2015

A230655 Squared radii of circles around a point of the hexagonal lattice that contain a record number of lattice points.

Original entry on oeis.org

0, 1, 7, 49, 91, 637, 1729, 8281, 12103, 53599, 157339, 375193, 1983163, 4877509, 13882141, 85276009, 180467833, 596932063, 3428888827, 4178524441, 7760116819, 29249671087, 36412855843, 147442219561, 254889990901, 473367125959, 1784229936307, 2439661341481

Offset: 1

Hugo Pfoertner, Oct 27 2013

nonn

It appears that this is also the sequence of numbers with a record number of divisors all of whose prime factors are of the form 3k + 1. - Amiram Eldar, Sep 12 2019 [This is correct, see A343771. - Jianing Song, May 19 2021]

Indices of records of A004016. Apart from the first term, also indices of records of A002324. - Jianing Song, May 20 2021

			a(2)=7 because a circle with radius sqrt(7) around the lattice point at (0,0) is the first circle that passes through more lattice points than a circle with radius 1, which passes through 6 points. The 12 hit points are (+-1/2,+-3*sqrt(3)/2), (+-2,+-sqrt(3)), (+-5/2, +-sqrt(3)/2).

Jianing Song, Table of n, a(n) for n = 1..247 (all terms <= 10^75.)

Cf. A004016, A002324.
Cf. A003136 (all occurring squared radii), A198799 (common terms), A230656 (index positions of records), A344472 (records).
Apart from the first term, subsequence of A343771.
Indices of records of Sum_{d|n} kronecker(m, d): this sequence (m=-3), A071383 (m=-4, similar sequence for square lattice), A279541 (m=-6).

my(v=list_A344473(10^15), rec=0); print1(0, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(v[n], ", "))) \\ Jianing Song, May 20 2021, see program for A344473

Offset corrected by Jianing Song, May 20 2021

A344472 Record values in A002324.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 108, 128, 144, 160, 192, 216, 256, 288, 320, 384, 432, 512, 576, 640, 768, 864, 1024, 1152, 1280, 1536, 1728, 2048, 2304, 2560, 3072, 3456, 3840, 4096, 4608, 5120, 5184, 6144, 6912, 7680, 8192, 9216

Offset: 1

Jianing Song, May 20 2021

nonn

Also numbers k such that A343771(m) > A343771(k) for all m > k.

			8 is a term because the circle with radius sqrt(1729) centered at the origin hits exactly 6*8 = 48 points in the A_2 lattice, and any circle with radius < sqrt(1729) centered at the origin hits fewer than 48 points.

Jianing Song, Table of n, a(n) for n = 1..246

Cf. A230655, A344471, A002324, A343771, A000005, A344473.

Records of Sum_{d|n} kronecker(m, d): this sequence (m=-3), A344470 (m=-4), A279542 (m=-6).

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

a(n) = A344471(n+1)/6.

a(n) = A000005(A230655(n+1)) = A002324(A230655(n+1)).

A344473 Numbers of the form (q1^b1)(q2^b2)(q3^b3)(q4^b4)(q5^b5)... where q1=7, q2=13, q3=19, q4=31, q5=37, ... (A002476) and b1>=b2>=b3>=b4>=b5...

Original entry on oeis.org

1, 7, 49, 91, 343, 637, 1729, 2401, 4459, 8281, 12103, 16807, 31213, 53599, 57967, 84721, 117649, 157339, 218491, 375193, 405769, 593047, 753571, 823543, 1101373, 1529437, 1983163, 2626351, 2840383, 2989441, 4151329, 4877509, 5274997, 5764801, 7709611

Offset: 1

Jianing Song, May 20 2021

A343771 is a subsequence.

			12103 is a term since 12103 = 7^2 * 13 * 19.
22477 is not a term since 22477 = 7 * 13^2 * 19, the exponents are not nonincreasing.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A002476, A343771, A054994.

\\ following program for A054994
list_A344473(lim) =
{
    my(u = [1], v = List(), w = v, pr, t = 1);
    forprime(p = 7, ,
        if (p % 3 > 1, next);
        t *= p;
        if (t > lim,
            break);
        listput(w, t)
    );
    for (i = 1, #w,
        pr = 1;
        for (e = 1, logint(lim\ = 1, w[i]),
            pr *= w[i];
            for (j = 1, #u,
                t = pr * u[j];
                if (t > lim,
                    break);
                listput(v, t)
            )
        );
        if (w[i] ^ 2 < lim, u = Set(concat(Vec(v), u)); v = List());
    );
    Set(concat(Vec(v), u));
}
list_A344473(100000)

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

Hugo Pfoertner, Sep 11 2022

nonn

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

Cf. A002324, A003136, A004016, A343771.

A374295 a(n) is the smallest positive integer k such that A096936(k) = n.

Original entry on oeis.org

1, 7, 4, 91, 2401, 28, 117649, 1729, 196, 31213, 282475249, 364, 13841287201, 1529437, 9604, 53599, 33232930569601, 2548, 1628413597910449, 593047, 470596, 3672178237, 3909821048582988049, 6916, 68574961, 179936733613, 33124, 29059303, 459986536544739960976801, 124852

Offset: 1

Seiichi Manyama, Jul 02 2024

nonn

a(n) is the smallest positive integer k such that A033716(k) = 2*n,

			   n        |        a(n)
------------+-------------------------------------
   2        |            7.
   3 = 3*1  |            4.
   4        |           91 =     7 * 13.
   5        |         2401 =     7^4.
   6 = 3*2  |           28 = 4 * 7.
   7        |       117649 =     7^6.
   8        |         1729 =     7 * 13 * 19.
   9 = 3*3  |          196 = 4 * 7^2.
  10        |        31213 =     7^4 * 13.
  11        |    282475249 =     7^10.
  12 = 3*4  |          364 = 4 * 7 * 13.
  13        |  13841287201 =     7^12.
  14        |      1529437 =     7^6 * 13.
  15 = 3*5  |         9604 = 4 * 7^4.
  16        |        53599 =     7 * 13 * 19 * 31.
  17        |                    7^16.
  18 = 3*6  |         2548 = 4 * 7^2 * 13.
  19        |                    7^18.
  20        |       593047 =     7^4 * 13 * 19.
  21 = 3*7  |       470596 = 4 * 7^6.
  22        |   3672178237 =     7^10 * 13.
  23        |                    7^22.
  24 = 3*8  |         6916 = 4 * 7 * 13 * 19.
  25        |     68574961 =     7^4 * 13^4.
  26        | 179936733613 =     7^12 * 13.
  27 = 3*9  |        33124 = 4 * 7^2 * 13^2.
  28        |     29059303 =     7^6 * 13 * 19.
  29        |                    7^28.
  30 = 3*10 |       124852 = 4 * 7^4 * 13.

Cf. A033716, A096936, A343771.

If p is prime, a(p) = 7^(p-1).

a(n) is divisible by 7 for n > 3.

cp's OEIS Frontend

A355919 Let n = p_1*p_2*...*p_k be the prime factorization of n, with the primes sorted in descending order; let b(n) = 7^(p_1 - 1)*13^(p_2 - 1)*19^(p_3 - 1)*...*A002476(k)^(p_k - 1). Sequence lists m such that b(m) > A343771(m).

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A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

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A230655 Squared radii of circles around a point of the hexagonal lattice that contain a record number of lattice points.

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PARI

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A344472 Record values in A002324.

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A344473 Numbers of the form (q1^b1)(q2^b2)(q3^b3)(q4^b4)(q5^b5)... where q1=7, q2=13, q3=19, q4=31, q5=37, ... (A002476) and b1>=b2>=b3>=b4>=b5...

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PARI

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

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A374295 a(n) is the smallest positive integer k such that A096936(k) = n.

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A355919 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order; let b(n) = 7^(p_1 - 1)13^(p_2 - 1)19^(p_3 - 1)...*A002476(k)^(p_k - 1). Sequence lists m such that b(m) > A343771(m).