A098098 -id:A098098 - cp's OEIS frontend

A383405 Partial sums of the sum of the divisors of the numbers of the form 6*k + 5, k >= 0.

6, 18, 36, 60, 90, 138, 180, 228, 282, 342, 426, 498, 594, 678, 768, 888, 990, 1098, 1212, 1356, 1512, 1644, 1782, 1950, 2100, 2292, 2484, 2652, 2826, 3006, 3234, 3426, 3624, 3864, 4104, 4368, 4620, 4848, 5082, 5322, 5664, 5916, 6174, 6438, 6708, 7080, 7362, 7698, 7992, 8328, 8700, 9012, 9330, 9690, 10074

Offset: 0

Omar E. Pol, Apr 25 2025

Consider a spiral similar to the spiral described in A239660 but instead of having four quadrants on the square grid the new spiral has six wedges on the triangular grid. A "diamond" formed by two adjacent triangles has area 1. a(n) is the total number of diamonds (or the total area) in the fifth wedge after n + 1 turns. The interesting fact is that for n >> 1 the geometric pattern in the fifth wedge of the spiral is very similar to the geometric pattern of the first wedge but it is different from the other wedges. Also the geometric pattern in the second wedge is very similar to the geometric pattern of the fourth wedge. Note that the six wedge spiral shows more and better geometric patterns than the four quadrants spiral.

The graph named W5 in the Plot 6 of the Links section is very close to the graph of A363161 (W1) and far from the graph of A365446 (W6).

OEIS Plot 2, Plot pairs of A363161 and A383405
Omar E. Pol, Plot 6. Area of the spiral in the six wedges

Sequences of the same family are A363161, A365442, A383403, A365444, this sequence, A365446.

Cf. A000203, A016969, A098098, A237593, A239660.

Accumulate@ Array[DivisorSigma[1, 6 # + 5] &, 55, 0] (* Michael De Vlieger, Apr 25 2025 *)

a(n) = sum(k=0, n, sigma(6*k+5)); \\ Michel Marcus, Apr 25 2025

a(n) = 6*Sum_{k=0..n} A098098(k).

a(n) = (Pi^2/3) * n^2 + O(n*log(n)). - Amiram Eldar, Apr 25 2025

A232343 Expansion of q^(-5/3) * c(q^2)^3 / (9 * c(q)) in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, 2, 0, 3, -2, 4, 0, 5, -5, 8, 0, 7, -4, 8, 0, 9, -8, 10, 0, 14, -6, 12, 0, 16, -14, 14, 0, 15, -8, 20, 0, 17, -14, 18, 0, 19, -10, 24, 0, 26, -21, 22, 0, 23, -16, 28, 0, 25, -20, 32, 0, 32, -14, 28, 0, 29, -28, 30, 0, 38, -16, 32, 0, 33, -31, 40, 0, 40

Offset: 0

Michael Somos, Nov 22 2013

sign

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

			G.f. = 1 - x + 2*x^2 + 3*x^4 - 2*x^5 + 4*x^6 + 5*x^8 - 5*x^9 + 8*x^10 + ...
G.f. = q^5 - q^8 + 2*q^11 + 3*q^17 - 2*q^20 + 4*q^23 + 5*q^29 - 5*q^32 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A033686, A098098.

Basis( ModularForms( Gamma0(18), 2), 210) [6]; /* Michael Somos, Jul 09 2018 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^6]^9 / (QPochhammer[ x^2] QPochhammer[ x^3])^3, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3] QPochhammer[ x^12])^3 / (QPochhammer[ -x] QPochhammer[ x^4]), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# == 2, 2 - 2^#2,# == 3, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[3 n + 5]) / 6]; (* Michael Somos, Jul 09 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^9 / (eta(x^2 + A) * eta(x^3 + A))^3, n))};

{a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 5; A = factor(n); 1/6 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 2 - 2^e, p==3, 0, (p^(e+1) - 1) / (p - 1))))};

Expansion of q^(-5/3) * eta(q) * eta(q^6)^9 / (eta(q^2) * eta(q^3))^3 in powers of q.
Euler transform of period 6 sequence [-1, 2, 2, 2, -1, -4, ...].
a(n) = 1/6 * b(3*n + 5) where b() is multiplicative with b(2^e) = 2 - 2^e, b(3^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
a(2*n) = A098098(n). a(4*n + 1) = - A033686(n). a(4*n + 3) = 0.

A232356 Expansion of 2/9 * c(q) * c(q^2) - q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function and c(q) is a cubic AGM theta function.

Original entry on oeis.org

1, 0, 5, -2, 6, 4, 8, -6, 17, 0, 12, 2, 14, 0, 30, -14, 18, 16, 20, -12, 40, 0, 24, -2, 31, 0, 53, -16, 30, 24, 32, -30, 60, 0, 48, 14, 38, 0, 70, -36, 42, 32, 44, -24, 102, 0, 48, -10, 57, 0, 90, -28, 54, 52, 72, -48, 100, 0, 60, 12, 62, 0, 136, -62, 84, 48

Offset: 1

Michael Somos, Nov 22 2013

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

			G.f. = q + 5*q^3 - 2*q^4 + 6*q^5 + 4*q^6 + 8*q^7 - 6*q^8 + 17*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008438, A033686, A098098, A111932, A121443, A123532, A144614, A229615, A232343.

Basis( ModularForms( Gamma0(6), 2), 70) [2];

a[ n_] := If[ n < 1, 0, Sum[ d ( 2 Mod[ d, 2] Boole[Mod[ n/d, 3] > 0] - Mod[ n/d, 2] Boole[ Mod[d, 3] > 0]), {d, Divisors @n}]];
a[ n_] := SeriesCoefficient[ 2 q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]) - q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}];

{a(n) = local(A); if( n<1, 0, n--; A=x*O(x^n); polcoeff( 2 * (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)) - (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};

ModularForms( Gamma0(6), 2, prec=70).1;

a(n) = 2 * A121443(n) - A111932(n). a(2*n) = -2 * A229615(n). a(12*n + 2) = a(12*n + 10) = 0.

a(n) = A123532(n) + 7 * A229615(n). a(3*n + 2) = 6 * A232343(n-1). a(6*n + 5) = 6 * A098098(n). a(12*n + 4) = -2 * A144614(n). a(12*n + 6) = 4 * A008438(n). a(12*n + 8) = -6 * A033686(n). - Michael Somos, May 23 2014

A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 2, 4, 4, 5, 6, 7, 10, 9, 12, 11, 14, 16, 14, 15, 16, 20, 22, 19, 20, 21, 22, 31, 28, 28, 26, 30, 34, 29, 30, 36, 32, 40, 38, 35, 36, 37, 56, 39, 40, 41, 42, 52, 48, 57, 50, 47, 62, 49, 50, 56, 60, 64, 54, 55, 62, 57, 70, 68, 60, 66, 62, 76, 70, 70, 76

Offset: 1

M. F. Hasler, Dec 08 2017

Robert G. Wilson v observes in A280098 that {1, 3, 4, 6, 8, 12, 24} seem to be the only positive integers k such that sigma(kn-1)/k is an integer for all n > 0.

Muniru A Asiru, Table of n, a(n) for n = 1..100000

Cf. A280098 (analog for k = 24), A097723 (analog for k = 4), A033686 (analog for k = 3), A000203 (sigma, also the  analog for k = 1).
The analog for k = 8 is A258835, up to the offset.
The analog for k = 6 is A098098 (up to the offset), a signed variant of this and the preceding one is A258831.
Cf. A086463.

sequence := List([1..10^5], n-> Sigma(12 *n-1)/12); # Muniru A Asiru, Dec 28 2017

with(numtheory):
seq(sigma(12*n-1)/12, n=1..10^3); # Muniru A Asiru, Dec 28 2017

Array[DivisorSigma[1, 12 # - 1]/12 &, 66] (* Michael De Vlieger, Dec 08 2017 *)

PARI
```
vector(90,n,sigma(12*n-1)/12)
```

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Mar 28 2024

A321528 Expansion of b(x)^2 * b(x^2) / b(x^4) where b is a cubic AGM theta function.

Original entry on oeis.org

1, -6, 6, 30, -66, -36, 186, -48, -210, 138, 36, -72, 114, -84, 48, 180, -498, -108, 726, -120, -396, 240, 72, -144, -30, -186, 84, 462, -528, -180, 1116, -192, -1074, 360, 108, -288, 654, -228, 120, 420, -1260, -252, 1488, -264, -792, 828, 144, -288, -318

Offset: 0

Michael Somos, Nov 12 2018

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 64 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			G.f. = 1 - 6*x + 6*x^2 + 30*x^3 - 66*x^4 - 36*x^5 + 186*x^6 - 48*x^7 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A098098, A134077, A321527.

A := Basis( ModularForms( Gamma0(12), 2), 49); A[1] - 6*A[2] + 6*A[3] + 30*A[4] - 66*A[5];

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^2])^3 / ( EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6]), {x, 0, n}];
a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n==0], -6 (s[n/1] - 4 s[n/2] - 9 s[n/3] + 16 s[n/4])]];
a[ n_] := If[ n < 1, Boole[n==0], -6 Sum[ d {1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0}[[Mod[d, 12, 1]]], {d, Divisors[n]}]];

{a(n) = if( n<1, n==0, -6 * sumdiv( n, d, d * [0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1][d%12 + 1]))};

{a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, -6 * (s(n/1) - 4*s(n/2) - 9*s(n/3) + 16*s(n/4)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^2 + A)^3 * eta(x^12 + A) / (eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};

Expansion of phi(-x) * phi(-x^2)^3 / (phi(-x^3) * phi(-x^6)) in powers of x where phi() is a Ramanujan theta function.
Expansion of eta(q)^6 * eta(q^2)^3 * eta(q^12) / (eta(q^3)^2 * eta(q^4)^3 * eta(q^6)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 864 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321527.
a(n) = -6 * (s(n/1) - 4*s(n/2) - 9*s(n/3) + 16*s(n/4)) if n>0, where s(x) = sum of divisors of x for integer x else 0.
a(2*n + 1) = -6 * A134077(n). a(6*n + 5) = -a(12*n + 10) = -36 * A098098(n).

A329651 Expansion of x * (psi(x^6) / psi(-x^3))^3 * phi(-x)^5 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

0, 1, -9, 31, -45, 6, 45, 8, -117, 121, -54, 12, 9, 14, -72, 186, -261, 18, 207, 20, -270, 248, -108, 24, -63, 31, -126, 391, -360, 30, 270, 32, -549, 372, -162, 48, 171, 38, -180, 434, -702, 42, 360, 44, -540, 726, -216, 48, -207, 57, -279, 558, -630, 54, 693

Offset: 0

Michael Somos, Nov 18 2019

sign

Number 105 of the 126 eta-quotients listed in Table 1 of Williams 2012.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 144 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A328788.

			G.f. = x - 9*x^2 + 31*x^3 - 45*x^4 + 6*x^5 + 45*x^6 + 8*x^7 - 117*x^8 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A000203, A098098, A133739, A134077, A328788.

A := Basis( ModularForms( Gamma0(12), 2), 52); A[2] - 9*A[3] + 31*A[4] - 45*A[5];

a[ n_] := SeriesCoefficient[ 1/2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, Pi/4, x^(3/2)])^3 EllipticTheta[ 4, 0, x]^5 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}] // PowerExpand;

{a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, 0, s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4))};

{a(n) = my(A); if( n < 1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^9 * eta(x^12 + A)^3 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^4 + A)), n))};

Euler transform of period 12 sequence [-9, -5, -6, -4, -9, -2, -9, -4, -6, -5, -9, -4, ...].
Expansion of x * phi(-x)^5 / psi(-x) * (f(-x^12) / f(-x^3))^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q)^9 * eta(q^12)^3 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^4)) in powers of q.
a(n) = s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4) if n>0 where s(x) = sum of divisors of x for integer x else 0.
a(n) = -(-1)^n * A133739(n). a(4*n + 2) = -9 * A134077(n). a(6*n + 5) = 6 * A098098(n).

A365412 a(n) = sigma(6n+2). Sum of the divisors of 6n+2, n >= 0.

Original entry on oeis.org

3, 15, 24, 42, 42, 63, 60, 84, 93, 120, 96, 126, 114, 186, 132, 168, 171, 210, 216, 210, 186, 255, 204, 336, 222, 300, 240, 294, 324, 372, 336, 336, 294, 465, 312, 378, 330, 504, 432, 420, 399, 480, 384, 588, 480, 558, 420, 504, 540, 570, 456, 672, 474, 762, 492, 588, 549, 660, 744

Offset: 0

Omar E. Pol, Sep 07 2023

Consider a spiral similar to the spiral described in A239660 but instead of having four quadrants on the square grid the new spiral has six wedges on the triangular grid. A "diamond" formed by two adjacent triangles has area 1. a(n) is the number of diamonds (or the area) added in the second wedge after n turns. The interesting fact is that for n >> 1 the geometric pattern in the second wedge of the spiral is similar to the geometric pattern of the fourth wedge but it is different from the other wedges.

Other members of the same family are A363031 and A224613. Also 6*A098098.
Partial sums give A365442.
Cf. A000203, A016933, A239660.

Table[DivisorSigma[1, 6*n + 2], {n, 0, 60}] (* Amiram Eldar, Sep 09 2023 *)

a(n) = A000203(6*n+2).

a(n) = A000203(A016933(n)).

A365414 a(n) = sigma(6n+4). Sum of the divisors of 6n+4, n >= 0.

Original entry on oeis.org

7, 18, 31, 36, 56, 54, 90, 72, 98, 90, 127, 144, 140, 126, 180, 144, 217, 162, 248, 180, 224, 252, 270, 216, 266, 288, 378, 252, 308, 270, 360, 360, 399, 306, 434, 324, 504, 342, 450, 432, 434, 468, 511, 396, 476, 414, 720, 504, 518, 450, 620, 576, 560, 576, 630, 504, 756, 522, 756, 540

Offset: 0

Omar E. Pol, Sep 07 2023

Consider a spiral similar to the spiral described in A239660 but instead of having four quadrants on the square grid the new spiral has six wedges on the triangular grid. A "diamond" formed by two adjacent triangles has area 1. a(n) is the number of diamonds (or the area) added in the fourth wedge after n turns. The interesting fact is that for n >> 1 the geometric pattern in the fourth wedge of the spiral is similar to the geometric pattern of the second wedge but it is different from the other wedges.

Partial sums give A365444.
Other members of the same family are A363031 and A224613. Also 6*A098098.
Cf. A000203, A016957, A363031.

Table[DivisorSigma[1, 6*n + 4], {n, 0, 60}] (* Amiram Eldar, Sep 09 2023 *)

a(n) = sigma(6*n+4); \\ Michel Marcus, Sep 08 2023

a(n) = A000203(6*n+4).

a(n) = A000203(A016957(n)).

cp's OEIS Frontend

A383405 Partial sums of the sum of the divisors of the numbers of the form 6*k + 5, k >= 0.

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A232343 Expansion of q^(-5/3) * c(q^2)^3 / (9 * c(q)) in powers of q where c() is a cubic AGM theta function.

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A232356 Expansion of 2/9 * c(q) * c(q^2) - q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function and c(q) is a cubic AGM theta function.

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A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

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A321528 Expansion of b(x)^2 * b(x^2) / b(x^4) where b is a cubic AGM theta function.

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A329651 Expansion of x * (psi(x^6) / psi(-x^3))^3 * phi(-x)^5 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

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A365412 a(n) = sigma(6n+2). Sum of the divisors of 6n+2, n >= 0.

A365414 a(n) = sigma(6n+4). Sum of the divisors of 6n+4, n >= 0.