A363031 -id:A363031 - 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

A072815 Sum of proper divisors of 6n + 1.

Original entry on oeis.org

0, 1, 1, 1, 6, 1, 1, 1, 8, 17, 1, 1, 1, 1, 23, 21, 1, 1, 1, 29, 12, 1, 27, 1, 35, 1, 1, 1, 14, 73, 1, 29, 1, 1, 47, 1, 39, 1, 1, 53, 1, 33, 35, 45, 59, 1, 1, 1, 18, 65, 51, 1, 1, 41, 109, 1, 1, 57, 1, 77, 20, 1, 1, 1, 191, 41, 1, 45, 1, 89, 1, 69, 1, 1, 95, 53, 1

Offset: 0

Hisanobu Shinya (ilikemathematics(AT)hotmail.com), Jul 14 2002

nonn

The square root of t(n) < s(t(4n-1, 4n-2, 4n-3, 4n-4)) < s(t(4n)).

			a(1) = s(t(1)) = 1 since t(1) = 7 and s(7) = 1 under the definition of the restricted divisor function.

T. D. Noe, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Restricted Divisor Function.

Cf. A001065, A016921, A145426, A363031.

Table[c=6n+1; DivisorSigma[1,c]-c, {n,0,80}] (* Harvey P. Dale, Nov 13 2013 *)

a(n) = sigma(6*n + 1) - 6*n - 1; \\ Amiram Eldar, Apr 12 2024

a(n) = s(t(n)), where t(n) = 6n + 1 and s(n) is the restricted divisor function.
From Amiram Eldar, Apr 12 2024: (Start)
a(n) = A363031(n) - A016921(n) = A001065(A016921(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = P^2/3 - 3 = A145426 = 0.289868... . (End)

Corrected by Harvey P. Dale, Nov 13 2013

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

A363161 Partial sums of A363031.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A072815 Sum of proper divisors of 6n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

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.