A294630 -id:A294630 - cp's OEIS frontend

A294628 a(n) = 8*(sigma(n) - n + (1/2)).

4, 12, 12, 28, 12, 52, 12, 60, 36, 68, 12, 132, 12, 84, 76, 124, 12, 172, 12, 180, 92, 116, 12, 292, 52, 132, 108, 228, 12, 340, 12, 252, 124, 164, 108, 444, 12, 180, 140, 404, 12, 436, 12, 324, 268, 212, 12, 612, 68, 348, 172, 372, 12, 532, 140, 516, 188, 260, 12, 868, 12, 276

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Partial sums give A294629.

Cf. A001065, A017113, A235796, A237593, A239050, A294015, A294016, A294017, A294630.

List([1..10^5],n->8*(Sigma(n)-n+(1/2))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sigma(8*n-1)/8, n=1..10^3); # Muniru A Asiru, Mar 04 2018

a[n_] := 8 (DivisorSigma[1, n] - n) + 4; Array[a, 62] (* Robert G. Wilson v, Dec 12 2017 *)

a(n) = 4*A294015(n).
a(n) = 8*(A001065(n) + (1/2)).
a(n) = 8*(A000203(n) - n + (1/2)).
a(n) = A239050(n) - 4*A235796(n).
a(n) = A017113(n-1) - 8*A235796(n).

A294629 Partial sums of A294628.

Original entry on oeis.org

4, 16, 28, 56, 68, 120, 132, 192, 228, 296, 308, 440, 452, 536, 612, 736, 748, 920, 932, 1112, 1204, 1320, 1332, 1624, 1676, 1808, 1916, 2144, 2156, 2496, 2508, 2760, 2884, 3048, 3156, 3600, 3612, 3792, 3932, 4336, 4348, 4784, 4796, 5120, 5388, 5600, 5612, 6224, 6292, 6640, 6812, 7184, 7196, 7728, 7868, 8384

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

a(n) is also the volume (and the number of cubes) in the n-th level (starting from the top) of the stepped pyramid described in A294630.

Number of terms less than 10^k, k=1,2,3,...: 1, 5, 19, 61, 195, 623, 1967, 6225, ... - Muniru A Asiru, Mar 04 2018

			Illustration of initial terms (n = 1..6):
.                                                  _ _ _ _ _ _
.                                _ _ _ _         _|     |     |_
.                _ _ _ _       _|   |   |_      |       |       |
.      _ _      |   |   |     |    _|_    |     |      _|_      |
.     |_|_|     |_ _|_ _|     |_ _|   |_ _|     |_ _ _|   |_ _ _|
.     |_|_|     |   |   |     |   |_ _|   |     |     |_ _|     |
.               |_ _|_ _|     |_    |    _|     |       |       |
.       4                       |_ _|_ _|       |_      |      _|
.                  16                             |_ _ _|_ _ _|
.                                  28
.                                                      56
.
.                                        _ _ _ _ _ _ _ _
.             _ _ _ _ _ _              _|       |       |_
.            |     |     |           _|         |         |_
.         _ _|     |     |_ _       |           |           |
.        |      _ _|_ _      |      |          _|_          |
.        |     |       |     |      |        _|   |_        |
.        |_ _ _|       |_ _ _|      |_ _ _ _|       |_ _ _ _|
.        |     |       |     |      |       |_     _|       |
.        |     |_ _ _ _|     |      |         |_ _|         |
.        |_ _      |      _ _|      |           |           |
.            |     |     |          |_          |          _|
.            |_ _ _|_ _ _|            |_        |        _|
.                                       |_ _ _ _|_ _ _ _|
.                 68
.                                              120
.
Note that for n >= 2 the structure has a hole (or hollow) in the center.
a(n) is the number of ON cells in the n-th diagram.

Iain Fox, Table of n, a(n) for n = 1..10000

For other related diagrams see A294630 (partial sums), A294016 and A237593.

Cf. A000203, A004125, A016742, A024916, A067436, A239050, A243980, A294015, A294017, A294628.

List([1..1000],n->Sum([1..n],k->8*(Sigma(k)-k+(1/2)))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sum(8*(sigma(k)-k+(1/2)),k=1..n),n=1..1000); # Muniru A Asiru, Mar 04 2018

f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@Array[f, 56] (* Robert G. Wilson v, Dec 12 2017 *)

a(n) = 4*(sum(k=1, n, n\k*k) - sum(k=2, n, n%k)) \\ Iain Fox, Dec 10 2017

first(n) = my(res = vector(n)); res[1] = 4; for(x=2, n, res[x] = res[x-1] + 8*(sigma(x) - x + (1/2))); res; \\ Iain Fox, Dec 10 2017

from math import isqrt
def A294629(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-n**2<<2 # Chai Wah Wu, Oct 22 2023

a(n) = 4*A294016(n).
a(n) = A016742(n) - 8*A004125(n).
a(n) = A016742(n) - 4*A067436(n).
a(n) = A243980(n) - 4*A004125(n).
a(n) = A243980(n) - 2*A067436(n).

cp's OEIS Frontend

A294628 a(n) = 8*(sigma(n) - n + (1/2)).

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