A189923 -id:A189923 - cp's OEIS frontend

A321222 a(n) = Sum_{d|n} mu(d)*d^n.

1, -3, -26, -15, -3124, 45864, -823542, -255, -19682, 9990233352, -285311670610, 2176246800, -302875106592252, 11111328602468784, 437893859848932344, -65535, -827240261886336764176, 101559568985784, -1978419655660313589123978, 99999904632567310800

Offset: 1

Ilya Gutkovskiy, Nov 06 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A008683, A023900, A046970, A063453, A067858, A189922, A189923.

Table[Sum[MoebiusMu[d] d^n, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[MoebiusMu[k] (k x)^k/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Product[1 - Boole[PrimeQ[d]] d^n, {d, Divisors[n]}], {n, 20}]

a(n) = sumdiv(n, d, moebius(d)*d^n) \\ Andrew Howroyd, Nov 06 2018

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

a(n) = Product_{p|n, p prime} (1 - p^n).

A322324 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Product_{p|n, p prime} (1 - p^k).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -3, -2, 0, 1, -7, -8, -1, 0, 1, -15, -26, -3, -4, 0, 1, -31, -80, -7, -24, 2, 0, 1, -63, -242, -15, -124, 24, -6, 0, 1, -127, -728, -31, -624, 182, -48, -1, 0, 1, -255, -2186, -63, -3124, 1200, -342, -3, -2, 0, 1, -511, -6560, -127, -15624, 7502, -2400, -7, -8, 4, 0

Offset: 1

Ilya Gutkovskiy, Dec 03 2018

			Square array begins:
  1,  1,   1,    1,     1,     1, ...
  0, -1,  -3,   -7,   -15,   -31, ...
  0, -2,  -8,  -26,   -80,  -242, ...
  0, -1,  -3,   -7,   -15,   -31, ...
  0, -4, -24, -124,  -624, -3124, ...
  0,  2,  24,  182,  1200,  7502, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..5 give A063524, A023900, A046970, A063453, A189922, A189923.

Cf. A008683, A059379, A059380, A321222 (diagonal).

Table[Function[k, Product[1 - Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j] j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[d] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

T(n, k) = sumdiv(n, d, moebius(d)*d^k);
matrix(6, 6, n, k, T(n, k-1)) \\ Michel Marcus, Dec 03 2018

G.f. of column k: Sum_{j>=1} mu(j)*j^k*x^j/(1 - x^j).
Dirichlet g.f. of column k: zeta(s)/zeta(s-k).
A(n,k) = Sum_{d|n} mu(d)*d^k.

cp's OEIS Frontend

A321222 a(n) = Sum_{d|n} mu(d)*d^n.

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