A299692 -id:A299692 - cp's OEIS frontend

A224880 a(n) = 2n + sum of divisors of n.

3, 7, 10, 15, 16, 24, 22, 31, 31, 38, 34, 52, 40, 52, 54, 63, 52, 75, 58, 82, 74, 80, 70, 108, 81, 94, 94, 112, 88, 132, 94, 127, 114, 122, 118, 163, 112, 136, 134, 170, 124, 180, 130, 172, 168, 164, 142, 220, 155, 193, 174, 202, 160, 228, 182, 232, 194, 206

Offset: 1

Hans Havermann, Jul 23 2013

nonn

This sequence is A033880 for the negative integers, thus making explicit the mapping noted in A075701.
From Omar E. Pol, Jun 21 2018: (Start)
a(n) is also the total area of the terraces and the vertical sides that are visible in the perspective view at the n-th level (starting from the top) of the stepped pyramid described in A245092.
Partial sums give A299692. (End)

			a(6) = 2*6 + (1+2+3+6) = 24.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A033879, A033880, A072691, A075701, A155085, A237593, A245092, A299692.

with(numtheory); seq(2*k+sigma(k),k=1..100); # Wesley Ivan Hurt, Jul 24 2013

Mathematica
```
Table[2*n+DivisorSigma[1,n],{n,64}]
```

vector(80, n, 2*n + sigma(n)) \\ Michel Marcus, Aug 19 2015

a(n) = A155085(n) + n.
a(n) = 2n + sigma(n) = A005843(n) + A000203(n) = A033879(n) + 2*A000203(n) = A033880(n) + 2*A005843(n) = 2*A155085(n) - A000203(n) = 2*A000203(n) - A033880(n). - Wesley Ivan Hurt, Jul 24 2013
G.f.: 2*x/(1 - x)^2 + Sum_{k>=1} x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 17 2017
a(n) = A001065(n) + A008585(n). - Omar E. Pol, Mar 06 2018
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = zeta(2)/2 + 1 = A072691 + 1 = 1.822467... . - Amiram Eldar, Mar 17 2024

A325300 a(n) is the number of faces of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

6, 9, 15, 20, 24, 31, 35, 42, 49, 59, 63, 72, 76, 84, 95, 106, 110, 121, 125

Offset: 1

Omar E. Pol, Apr 16 2019

To calculate a(n) consider that levels greater than n do not exist.
The shape of the n-th level of the pyramid allows us to know if n is prime (see the Formula section).
For more information about the sequences that we can see in the pyramid see A262626.

			For n = 1 the first level of the stepped pyramid (starting from the top) is a cube, and a cube has six faces, so a(1) = 6.

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A325301 (number of edges), A325302 (number of vertices).

Cf. A196020, A215848, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A299692, A323645, A323648.

a(n) = A325301(n) - A325302(n) + 2 (Euler's formula).
a(n) = A323645(n) + 3.
a(n) = a(n-1) + 4 iff n is a prime > 3 (A215848).

A328366 a(n) is the surface area of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

6, 20, 40, 70, 102, 150, 194, 256, 318, 394, 462, 566, 646, 750, 858, 984, 1088, 1238, 1354, 1518, 1666, 1826, 1966, 2182, 2344, 2532, 2720, 2944, 3120, 3384, 3572, 3826, 4054, 4298, 4534, 4860, 5084, 5356, 5624, 5964, 6212, 6572, 6832, 7176, 7512, 7840, 8124, 8564, 8874, 9260, 9608, 10012

Offset: 1

Omar E. Pol, Oct 26 2019

nonn

			For n = 1 the first level of the stepped pyramid is a cube, so a(1) = 6.

Cf. A000217, A002378, A013661, A024916, A045943, A046092, A153485, A175254 (volume), A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A299692, A327329.

s=0;Do[s=s+4*DivisorSigma[1,n];t=2n(n+1);Print[(s/2)+t],{n,1,8000}] (* Metin Sariyar, Nov 20 2019 *)

from math import isqrt
def A328366(n): return (n*(n+1)<<1)-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 22 2023

a(n) = 4*A000217(n) + 2*A024916(n).
a(n) = 2*(A002378(n) + A327329(n)).
a(n) = 2*(A045943(n) + A153485(n)).
a(n) = A046092(n) + A327329(n).
a(n) = 2*A299692(n).
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2) + 2 = 3.644934... . - Amiram Eldar, Mar 21 2024

A323645 a(n) is the number of visible faces in the perspective view of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

3, 6, 12, 17, 21, 28, 32, 39, 46, 56, 60, 69, 73, 81, 92, 103, 107, 118, 122

Offset: 1

Omar E. Pol, Mar 20 2019

The shape of the n-th level of the pyramid allows us to know if n is prime (see the Formula section).

For more sequences that we can find in the pyramid see A262626.

Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16) and A000203

Cf. A000040, A000203, A024916, A196020, A215848, A224880, A235791, A236104, A237048, A237270, A237271, A237593, A245092, A262626, A299692 (total area).

a(n) = a(n-1) + 4 iff n is a prime > 3 (A215848).

a(n) = A325300(n) - 3. - Omar E. Pol, Apr 17 2019

a(18)-a(19) from Omar E. Pol, Apr 18 2019

cp's OEIS Frontend

A224880 a(n) = 2n + sum of divisors of n.

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A325300 a(n) is the number of faces of the stepped pyramid with n levels described in A245092.

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A328366 a(n) is the surface area of the stepped pyramid with n levels described in A245092.

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Mathematica

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A323645 a(n) is the number of visible faces in the perspective view of the stepped pyramid with n levels described in A245092.

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