A078471 -id:A078471 - cp's OEIS frontend

A366723 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+4,5).

1, 5, 21, 50, 121, 236, 447, 736, 1247, 1896, 2898, 4151, 5972, 8146, 11292, 14797, 19643, 25248, 32564, 40663, 51515, 63168, 78119, 94452, 114998, 136933, 164849, 193753, 229714, 268334, 314711, 362824, 422746, 483950, 558046, 635070, 726461, 820420, 934186, 1048245

Offset: 1

Seiichi Manyama, Oct 24 2023

nonn

Partial sums of A366814.
Cf. A078471, A366395, A366659.
Cf. A365439.

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(n\k+4, 5));

a(n) = Sum_{k=1..n} binomial(k+3,4) * (floor(n/k) mod 2).

G.f.: -1/(1-x) * Sum_{k>=1} (-x)^k/(1-x^k)^5 = 1/(1-x) * Sum_{k>=1} binomial(k+3,4) * x^k/(1+x^k).

A078472 Partial sums of A035282.

Original entry on oeis.org

1, 6, 12, 22, 46, 67, 107, 137, 168, 228, 292, 342, 426, 546, 606, 656, 800, 920, 1044, 1129, 1273, 1473, 1633, 1759, 1850, 2030, 2270, 2510, 2665, 2869, 3089, 3389, 3799, 4119, 4275, 4539, 4819, 5029, 5389, 5689, 5993, 6377, 6797, 6967, 7367, 7871, 8231

Offset: 1

Benoit Cloitre, Dec 31 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A078471, A035282.

Distinct terms of A353997.

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Select[Array[s, 200], # > 0 &]] (* Amiram Eldar, May 13 2022 *)

a(20) and a(36) corrected by Georg Fischer, Aug 31 2020

A309124 a(n) = n - 3 * floor(n/3) + 5 * floor(n/5) - 7 * floor(n/7) + ...

Original entry on oeis.org

1, 2, 0, 1, 7, 5, -1, 0, 7, 13, 3, 1, 15, 9, -3, -2, 16, 23, 5, 11, 23, 13, -9, -11, 20, 34, 14, 8, 38, 26, -4, -3, 17, 35, -1, 6, 44, 26, -2, 4, 46, 58, 16, 6, 48, 26, -20, -22, 21, 52, 16, 30, 84, 64, 4, -2, 34, 64, 6, -6, 56, 26, -16, -15, 69, 89, 23, 41, 85, 49, -21, -14, 60, 98, 36

Offset: 1

Ilya Gutkovskiy, Jul 13 2019

sign

Partial sums of A050457.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014200, A024919, A050457, A078471.

f:= proc(n) local r,d;
  r:= n/2^padic:-ordp(n,2);
  add((-1)^((d-1)/2)*d, d = numtheory:-divisors(r))
end proc:
ListTools:-PartialSums(map(f,[$1..100])); # Robert Israel, Oct 28 2020

Table[Sum[(-1)^(k + 1) (2 k - 1) Floor[n/(2 k - 1)], {k, 1, n}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[1/(1 - x) Sum[(-1)^(k + 1) (2 k - 1) x^(2 k - 1)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: (1/(1 - x)) * Sum_{k>=1} (-1)^(k+1) * (2*k - 1) * x^(2*k-1)/(1 - x^(2*k-1)).

A024930 a(n) = sum of remainders of n mod 1,3,5,...,2k-1, where k = [ (n+1)/2 ].

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 3, 6, 6, 5, 9, 11, 16, 15, 13, 20, 27, 23, 31, 35, 34, 33, 43, 51, 57, 56, 56, 62, 75, 66, 80, 95, 96, 95, 99, 104, 121, 120, 122, 136, 155, 144, 164, 174, 163, 162, 184, 204, 220, 214, 218, 230, 255, 242, 252, 272, 277, 276, 304, 310, 339, 338, 328, 359, 372, 357

Offset: 1

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A078471, A093005.

A093005:= [seq(n*ceil(n/2),n=1..100)]:
A078471:= ListTools:-PartialSums([seq(numtheory:-sigma(n/2^padic:-ordp(n,2)),n=1..100)]):
A093005-A078471; # Robert Israel, May 13 2019

a(n) = A093005(n) - A078471(n). - Robert Israel, May 13 2019

A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 11, 11, 19, 16, 21, 21, 30, 30, 37, 29, 45, 45, 51, 51, 66, 56, 67, 67, 88, 83, 96, 84, 105, 105, 112, 112, 144, 130, 147, 135, 159, 159, 178, 162, 197, 197, 208, 208, 241, 209, 232, 232, 277, 270, 290, 270, 309, 309, 324, 308, 357, 335, 364, 364

Offset: 1

Ilya Gutkovskiy, May 25 2020

nonn

Cf. A000217, A000593, A078471, A120885, A330926, A332490.

b:= n-> add(d, d=select(x-> x::odd, numtheory[divisors](n))):
a:= proc(n) option remember; n+`if`(n<2, 0, a(n-1))-b(n-1) end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 25 2020

Table[Sum[Mod[Ceiling[n/k], 2] k, {k, 1, n}], {n, 1, 60}]
Table[n (n + 1)/2 - Sum[DivisorSum[k, (-1)^(k/# + 1) # &], {k, 1, n - 1}], {n, 1, 60}]
nmax = 60; CoefficientList[Series[x/(1 - x) (1/(1 - x)^2 - Sum[k x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, (ceil(n/k) % 2)*k); \\ Michel Marcus, May 26 2020

G.f.: (x/(1 - x)) * (1/(1 - x)^2 - Sum_{k>=1} k * x^k / (1 + x^k)).
a(n) = n*(n + 1)/2 - Sum_{k=1..n-1} A000593(k).
a(n) = A000217(n) - A078471(n-1).

A262535 Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A261699.

Original entry on oeis.org

1, 2, 3, 3, 4, 3, 5, 8, 6, 8, 3, 7, 15, 3, 8, 15, 3, 9, 24, 6, 10, 24, 6, 5, 11, 35, 6, 5, 12, 35, 9, 5, 13, 48, 9, 5, 14, 48, 9, 12, 15, 63, 12, 12, 5, 16, 63, 12, 12, 5, 17, 80, 12, 12, 5, 18, 80, 15, 21, 5, 19, 99, 15, 21, 5, 20, 99, 15, 21, 10, 21, 120, 18, 21, 10, 7, 22, 120, 18, 32, 10, 7, 23, 143, 18, 32, 10, 7, 24, 143, 21, 32, 10, 7, 25, 168, 21, 32, 15, 7, 26, 168, 21, 45, 15, 7, 27, 195, 24, 45, 15, 16

Offset: 1

Omar E. Pol, Sep 24 2015

Conjecture: the sum of row n gives A078471(n), the sum of all odd divisors of all positive integers <= n.
Row n has length A003056(n) hence column k starts in row A000217(k).
Column 1 gives A000027.

			Triangle begins:
1;
2;
3,    3;
4,    3;
5,    8;
6,    8,  3;
7,   15,  3;
8,   15,  3;
9,   24,  6;
10,  24,  6,  5;
11,  35,  6,  5;
12,  35,  9,  5;
13,  48,  9,  5;
14,  48,  9, 12;
15,  63, 12, 12,  5;
16,  63, 12, 12,  5;
17,  80, 12, 12,  5;
18,  80, 15, 21,  5;
19,  99, 15, 21,  5;
20,  99, 15, 21, 10;
21, 120, 18, 21, 10,  7;
22, 120, 18, 32, 10,  7;
23, 143, 18, 32, 10,  7;
24, 143, 21, 32, 10,  7;
25, 168, 21, 32, 15,  7;
26, 168, 21, 45, 15,  7;
27, 195, 24, 45, 15, 16;
...
For n = 6 the sum of all odd divisors of all positive integers <= 6 is (1) + (1) + (1 + 3) + (1) + (1 + 5) + (1 + 3) = 17. On the other hand the sum of the 6th row of triangle is 6 + 8 + 3 = 17 equaling the sum of all odd divisors of all positive integers <= 6.

Cf. A000027, A000217, A001227, A003056, A060831, A078471, A236104, A237593, A261699.

cp's OEIS Frontend

A366723 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+4,5).

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A078472 Partial sums of A035282.

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A309124 a(n) = n - 3 * floor(n/3) + 5 * floor(n/5) - 7 * floor(n/7) + ...

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A024930 a(n) = sum of remainders of n mod 1,3,5,...,2k-1, where k = [ (n+1)/2 ].

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A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.

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A262535 Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A261699.

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