A383403 -id:A383403 - cp's OEIS frontend

A365442 Partial sums of A365412.

3, 18, 42, 84, 126, 189, 249, 333, 426, 546, 642, 768, 882, 1068, 1200, 1368, 1539, 1749, 1965, 2175, 2361, 2616, 2820, 3156, 3378, 3678, 3918, 4212, 4536, 4908, 5244, 5580, 5874, 6339, 6651, 7029, 7359, 7863, 8295, 8715, 9114, 9594, 9978, 10566, 11046, 11604, 12024, 12528

Offset: 0

Omar E. Pol, Sep 07 2023

Partial sums of the sum of the divisors of the numbers of the form 6*k + 2, 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 second wedge after n + 1 turns. The interesting fact is that for n >> 1 the geometric pattern in the second wedge of the spiral is very similar to the geometric pattern of the fourth wedge but it is different from the other wedges. Note that the six wedge spiral shows more and better geometric patterns than the four quadrants spiral.
The graph is very close to the graph of A365444 (see the Links section).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
OEIS Plot 2, Plot pairs of A365442 and A365444
Omar E. Pol, Plot 6. Area of the spiral in the six wedges

Sequences of the same family are A363161, this sequence, A383403, A365444, A383405, A365446.

Cf. A000203, A016933, A237593, A239660, A365412.

Accumulate[Table[DivisorSigma[1, 6*n + 2], {n, 0, 50}]] (* Amiram Eldar, Sep 08 2023 *)

a(n) = sum(k=0, n, sigma(6*k+2)); \\ Michel Marcus, Sep 09 2023

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

cp's OEIS Frontend

A365442 Partial sums of A365412.

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