A365442 -id:A365442 - cp's OEIS frontend

A363161 Partial sums of A363031.

1, 9, 23, 43, 74, 106, 144, 188, 245, 317, 379, 447, 521, 601, 709, 821, 919, 1023, 1133, 1277, 1410, 1538, 1698, 1838, 2018, 2170, 2328, 2492, 2675, 2923, 3105, 3321, 3515, 3715, 3967, 4179, 4435, 4659, 4889, 5177, 5419, 5699, 5987, 6291, 6615, 6887, 7165, 7449, 7756, 8116, 8468, 8776, 9090, 9450, 9884

Offset: 0

Omar E. Pol, May 18 2023

Partial sums of the sum of the divisors of the numbers of the form 6*k + 1, k >= 0.
From Omar E. Pol, Sep 26 2023: (Start)
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 first wedge after n turns. The interesting fact is that for n >> 1 the geometric pattern in the first wedge is similar to the geometric pattern of the fifth wedge but it is different from the other wedges. (End)

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Omar E. Pol, Plot 6. Area of the spiral in the six wedges after n turns

Other sequences of the same family are A365442, A365444, A365446.

Cf. A016921, A239660, A363031.

Accumulate@ Array[DivisorSigma[1, 6 # + 1] &, 55, 0] (* Michael De Vlieger, Aug 27 2023 *)

a(n) = sum(k=0, n, sigma(6*k+1)); \\ Michel Marcus, Aug 28 2023

a(n) ~ Pi^2 * n^2 / 3. - Vaclav Kotesovec, Aug 31 2023

A365444 Partial sums of A365414.

Original entry on oeis.org

7, 25, 56, 92, 148, 202, 292, 364, 462, 552, 679, 823, 963, 1089, 1269, 1413, 1630, 1792, 2040, 2220, 2444, 2696, 2966, 3182, 3448, 3736, 4114, 4366, 4674, 4944, 5304, 5664, 6063, 6369, 6803, 7127, 7631, 7973, 8423, 8855, 9289, 9757, 10268, 10664, 11140, 11554, 12274, 12778

Offset: 0

Omar E. Pol, Sep 07 2023

Partial sums of the sum of the divisors of the numbers of the form 6*k + 4, k >= 0.

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 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.

OEIS Plot 2, Plot pairs of A365444 and A365442.
Omar E. Pol, Plot 6. Area of the spiral in the six wedges after n turns

Other sequences of the same family are A363161, A365442, A365446.

Cf. A000203, A016957, A239660, A365414.

Accumulate[Table[DivisorSigma[1, 6*n + 4], {n, 0, 50}]] (* Amiram Eldar, Sep 08 2023 *)

a(n) = sum(k=0, n, sigma(6*k+4)); \\ Michel Marcus, Sep 08 2023

a(n) = (5*Pi^2/9) * n^2 + O(n*log(n)). - Amiram Eldar, Sep 08 2023

A365446 Partial sums of A224613.

Original entry on oeis.org

12, 40, 79, 139, 211, 302, 398, 522, 642, 810, 954, 1149, 1317, 1541, 1775, 2027, 2243, 2523, 2763, 3123, 3435, 3771, 4059, 4462, 4834, 5226, 5589, 6069, 6429, 6975, 7359, 7867, 8335, 8839, 9415, 10015, 10471, 11031, 11577, 12321, 12825, 13553, 14081, 14801, 15521, 16193, 16769, 17588, 18272, 19140, 19842

Offset: 1

Omar E. Pol, Sep 04 2023

Partial sums of the sum of the divisors of the nonzero multiples of 6.

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 sixth wedge after n turns. Note that the six wedge spiral shows more and better geometric patterns than the four quadrants spiral. - Omar E. Pol, Apr 26 2025

Omar E. Pol, Plot 6. Area of the spiral in the six wedges

Sequences of the same family are A363161, A365442, A365444, A383405, this sequence.

Cf. A000203, A008588, A224613, A237593, A239660.

Accumulate[Table[DivisorSigma[1, 6*n], {n, 1, 50}]] (* Amiram Eldar, Sep 07 2023 *)

from math import prod
from collections import Counter
from sympy import factorint
def A365446(n): return sum(prod((p**(e+1)-1)//(p-1) for p, e in (Counter(factorint(m))+Counter([2,3])).items()) for m in range(1,n+1)) # Chai Wah Wu, Sep 07 2023

a(n) = (55*Pi^2/72) * n^2 + O(n*log(n)). - Amiram Eldar, Sep 07 2023

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

Original entry on oeis.org

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

A383403 Partial sums of the sum of the divisors of the numbers of the form 6*k + 3, k >= 0.

Original entry on oeis.org

4, 17, 41, 73, 113, 161, 217, 295, 367, 447, 551, 647, 771, 892, 1012, 1140, 1296, 1488, 1640, 1822, 1990, 2166, 2406, 2598, 2826, 3060, 3276, 3564, 3824, 4064, 4312, 4632, 4968, 5240, 5552, 5840, 6136, 6539, 6923, 7243, 7607, 7943, 8375, 8765, 9125, 9573, 9989, 10469, 10861

Offset: 0

Omar E. Pol, Apr 27 2025

Partial sums of the sum of the divisors of A016945.
See the illustration of a(3) and a(10) as the total area (or total number of cells) in the diagram of the symmetric representation of sigma in the Links section.
Also consider a spiral similar to the spiral described in A239660 but with 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 third wedge after n + 1 turns. The spiral can be visualized from the top view of the stepped pyramid described in A274536. The graph is named W3 in the Plot 6 of the Links section.

			For n = 3 the first four terms of the numbers of the form 6*k + 3, k >= 0, are [3, 9, 15, 21]. The divisors of them are [1, 3], [1, 3, 9], [1, 3, 5, 15], [1, 3, 7, 21]. The sum of the divisors of them are [4, 13, 24, 32] respectively, and the sum of all divisors of them are 4 + 13 + 24 + 32 = 73, so a(3) = 73.

Omar E. Pol, Illustration of a(3) = 73
Omar E. Pol, Illustration of a(10) = 551
Omar E. Pol, Plot 6. Area of the spiral in the six wedges

Sequences of the same family are A363161, A365442, this sequence, A365444, A383405, A365446.

Cf. A000203, A016945, A237593, A239660, A274536.

Accumulate@ Array[DivisorSigma[1, 6 # + 3] &, 55, 0]

PARI
```
a(n) = sum(k=0, n, sigma(6*k+3));
```

a(n) = Sum_{k=0..n} sigma(6*k+3).

a(n) = (11*Pi^2/24) * n^2 + O(n*log(n)). - Amiram Eldar, Apr 28 2025

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)).

cp's OEIS Frontend

A363161 Partial sums of A363031.

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A365444 Partial sums of A365414.

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A365446 Partial sums of A224613.

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

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A383403 Partial sums of the sum of the divisors of the numbers of the form 6*k + 3, k >= 0.

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

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