A321438 -id:A321438 - cp's OEIS frontend

A076717 a(n) = -Sum_{d|n} (-n/d)^d.

1, 1, 4, -1, 6, 4, 8, -25, 37, 16, 12, -106, 14, 92, 384, -561, 18, -65, 20, -706, 2552, 1948, 24, -15658, 3151, 8048, 20440, -2570, 30, -33326, 32, -135393, 178512, 130816, 94968, -583219, 38, 523964, 1596560, -2465370, 42, -2521186, 44, -15082, 16364502, 8388124, 48, -78560082, 823593, 23888231

Offset: 1

Vladeta Jovovic, Oct 27 2002

sign

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A055225, A308367, A321438.

a(n) = -sumdiv(n, d, (-n/d)^d); \\ Michel Marcus, Mar 22 2021

G.f.: Sum_{n>0} n*x^n/(1+n*x^n).

A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 3, 4, -1, 1, 7, 10, 1, 2, 1, 15, 28, 11, 6, 0, 1, 31, 82, 55, 26, 4, 2, 1, 63, 244, 239, 126, 30, 8, -2, 1, 127, 730, 991, 626, 196, 50, 1, 3, 1, 255, 2188, 4031, 3126, 1230, 344, 43, 13, 0, 1, 511, 6562, 16255, 15626, 7564, 2402, 439, 91, 6, 2, 1, 1023, 19684, 65279, 78126, 45990, 16808, 3823, 757, 78, 12, -2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
   1,  1,   1,    1,     1,     1,  ...
   0,  1,   3,    7,    15,    31,  ...
   2,  4,  10,   28,    82,   244,  ...
  -1,  1,  11,   55,   239,   991,  ...
   2,  6,  26,  126,   626,  3126,  ...
   0,  4,  30,  196,  1230,  7564,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A048272, A000593, A078306, A078307, A284900, A284926, A284927, A321552, A321553, A321554, A321555, A321556, A321557.

Cf. A109974, A279394, A279396, A285425, A321438 (diagonal), A322082, A322083, A322084.

Table[Function[k, Sum[(-1)^(n/d + 1) d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, (-1)^(n/d+1)*d^k)}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} j^k*x^j/(1 + x^j).

A342828 a(n) = Sum_{d|n} (-1)^(n/d+1) * d^(n-d).

Original entry on oeis.org

1, 0, 2, -4, 2, -11, 2, -320, 731, -2869, 2, -1827, 2, -819447, 10297068, -33570816, 2, 1775078476, 2, -36222872973, 678610493340, -285310622035, 2, 169888943418701, 95367431640627, -302875089815037, 150094917726535604, -569376395999240231, 2, 104002456598734754865, 2

Offset: 1

Seiichi Manyama, Mar 23 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..599

Cf. A308367, A321438, A342628.

a[n_] := DivisorSum[n, (-1)^(n/# + 1) * #^(n - #) &]; Array[a, 30] (* Amiram Eldar, Mar 23 2021 *)

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^(n-d));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+(k*x)^k)))

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

If p is an odd prime, a(p) = 2.

cp's OEIS Frontend

A076717 a(n) = -Sum_{d|n} (-n/d)^d.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A342828 a(n) = Sum_{d|n} (-1)^(n/d+1) * d^(n-d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula