A228441 -id:A228441

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A322083

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

+10
16

1, 1, -2, 1, -3, 2, 1, -5, 4, -1, 1, -9, 10, -3, 2, 1, -17, 28, -13, 6, -4, 1, -33, 82, -57, 26, -12, 2, 1, -65, 244, -241, 126, -50, 8, 0, 1, -129, 730, -993, 626, -252, 50, -3, 3, 1, -257, 2188, -4033, 3126, -1394, 344, -45, 13, -4, 1, -513, 6562, -16257, 15626, -8052, 2402, -441, 91, -18, 2

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

For each k, the k-th column sequence (T(n,k))(n>=1) is a multiplicative function of n, equal to (-1)^(n+1)*(Id_k * 1) in the notation of the Bala link. - Peter Bala, Mar 19 2022

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Peter Bala, A signed Dirichlet product of arithmetical functions

Index entries for sequences mentioned by Glaisher

FORMULA

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

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, ...

-2, -3, -5, -9, -17, -33, ...

2, 4, 10, 28, 82, 244, ...

-1, -3, -13, -57, -241, -993, ...

2, 6, 26, 126, 626, 3126, ...

-4, -12, -50, -252, -1394, -8052, ...

MATHEMATICA

Table[Function[k, Sum[(-1)^(n/d+d) d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

Table[Function[k, SeriesCoefficient[Sum[(-1)^(j + 1) j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

f[p_, e_, k_] := If[k == 0, e + 1, (p^(k*e + k) - 1)/(p^k - 1)]; f[2, e_, k_] := If[k == 0, e - 3, -((2^(k - 1) - 1)*2^(k*e + 1) + 2^(k + 1) - 1)/(2^k - 1)]; T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k, k], {n, 1, 11}, {k, n - 1, 0, -1}] // Flatten (* Amiram Eldar, Nov 22 2022 *)

PROG

(PARI) T(n, k)={sumdiv(n, d, (-1)^(n/d+d)*d^k)}

for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

CROSSREFS

Columns k=0..12 give A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809.

Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322084.

KEYWORD

sign,tabl

AUTHOR

Ilya Gutkovskiy, Nov 26 2018

STATUS

approved

A321558

a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^2.

+10
7

1, -5, 10, -13, 26, -50, 50, -45, 91, -130, 122, -130, 170, -250, 260, -173, 290, -455, 362, -338, 500, -610, 530, -450, 651, -850, 820, -650, 842, -1300, 962, -685, 1220, -1450, 1300, -1183, 1370, -1810, 1700, -1170, 1682, -2500, 1850, -1586, 2366

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

Peter Bala, A signed Dirichlet product of arithmetical functions

J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

Index entries for sequences mentioned by Glaisher

FORMULA

G.f.: Sum_{k>=1} (-1)^(k+1)*k^2*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

G.f.: Sum_{k>=1} (-1)^(k+1)*(x^k - x^(2*k))/(1 + x^k)^3. - Michael Somos, Oct 24 2019

a(n) = -(-1)^n A328667(n). a(2*n + 1) = A078306(2*n + 1). a(2*n) = A078306(2*n) - 8*A078306(n). - Michael Somos, Oct 24 2019

From Peter Bala, Jan 29 2022: (Start)

Multiplicative with a(2^k) = - (2^(2*k+1) + 7)/3 for k >= 1 and a(p^k) = (p^(2*k+2) - 1)/(p^2 - 1) for odd prime p.

n^2 = (-1)^(n+1)*Sum_{d divides n} A067856(n/d)*a(d). (End)

EXAMPLE

G.f. = x - 5*x^2 + 10*x^3 - 13*x^4 + 26*x^5 - 50*x^6 + 50*x^7 + ... - Michael Somos, Oct 24 2019

MATHEMATICA

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^2 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

PROG

(PARI) apply( A321558(n)=sumdiv(n, d, (-1)^(n\d-d)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018

(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^2*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

(Sage) s=(sum((-1)^(k+1)*k^2*x^k/(1 + x^k) for k in (1..50))).series(x, 30); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

CROSSREFS

Column k=2 of A322083.

Glaisher's xi_i (i=0..12): A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809

Cf. A321543 - A321557, A321810 - A321836 for similar sequences.

Cf. A078306, A328667.

KEYWORD

sign,mult,look

AUTHOR

N. J. A. Sloane, Nov 23 2018

STATUS

approved

A348951

a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d).

+10
5

0, 1, -1, 1, -1, 0, -1, 2, -1, 0, -1, 3, -1, 0, -2, 2, -1, 1, -1, 1, -2, 0, -1, 4, -1, 0, -2, 1, -1, 2, -1, 3, -2, 0, -2, 2, -1, 0, -2, 4, -1, 0, -1, 1, -3, 0, -1, 5, -1, 1, -2, 1, -1, 0, -2, 4, -2, 0, -1, 4, -1, 0, -3, 3, -2, 0, -1, 1, -2, 2, -1, 4, -1, 0, -3, 1, -2, 0, -1, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

FORMULA

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

MATHEMATICA

Table[DivisorSum[n, (-1)^(n/#) &, # < Sqrt[n] &], {n, 1, 80}]

nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PROG

(PARI) A348951(n) = sumdiv(n, d, if((d*d)<n, (-1)^(n/d), 0)); \\ Antti Karttunen, Nov 05 2021

CROSSREFS

Cf. A048272, A056924, A113652, A228441, A333809, A348515, A348952, A348953, A348954, A348955, A348956.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Nov 04 2021

STATUS

approved

A348952

a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d).

+10
5

0, 1, -1, 1, -1, 2, -1, 0, -1, 2, -1, 1, -1, 2, -2, 0, -1, 3, -1, 1, -2, 2, -1, 0, -1, 2, -2, 1, -1, 4, -1, -1, -2, 2, -2, 2, -1, 2, -2, 0, -1, 4, -1, 1, -3, 2, -1, -1, -1, 3, -2, 1, -1, 4, -2, 0, -2, 2, -1, 2, -1, 2, -3, -1, -2, 4, -1, 1, -2, 4, -1, 0, -1, 2, -3, 1, -2, 4, -1, -1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

FORMULA

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

For p odd prime, a(p) = a(p^2) = -1. - Bernard Schott, Nov 22 2021

MATHEMATICA

Table[-DivisorSum[n, (-1)^(# + n/#) &, # < Sqrt[n] &], {n, 1, 80}]

nmax = 80; CoefficientList[Series[Sum[x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PROG

(PARI) A348952(n) = -sumdiv(n, d, if((d*d)<n, (-1)^(d + (n/d)), 0)); \\ Antti Karttunen, Nov 05 2021

CROSSREFS

Cf. A048272, A056924, A228441, A258453, A305152, A333809, A348515, A348951, A348953, A348954, A348955, A348956.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Nov 04 2021

STATUS

approved

A348515

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

+10
3

1, -1, 1, -2, 1, 0, 1, -2, 2, 0, 1, -3, 1, 0, 2, -3, 1, -1, 1, -1, 2, 0, 1, -4, 2, 0, 2, -1, 1, -2, 1, -3, 2, 0, 2, -3, 1, 0, 2, -4, 1, 0, 1, -1, 3, 0, 1, -5, 2, -1, 2, -1, 1, 0, 2, -4, 2, 0, 1, -4, 1, 0, 3, -4, 2, 0, 1, -1, 2, -2, 1, -4, 1, 0, 3, -1, 2, 0, 1, -5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Michel Marcus, Table of n, a(n) for n = 1..10000

FORMULA

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

a(n) = 1 iff n = 1 or n is an odd prime (A006005). - Bernard Schott, Nov 22 2021

MATHEMATICA

Table[DivisorSum[n, (-1)^(n/# + 1) &, # <= Sqrt[n] &], {n, 1, 80}]

nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PROG

(PARI) A348515(n) = sumdiv(n, d, if((d*d)<=n, (-1)^(1 + (n/d)), 0)); \\ Antti Karttunen, Nov 05 2021

(Python)

from sympy import divisors

def a(n): return sum((-1)**(n//d + 1) for d in divisors(n) if d*d <= n)

print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 22 2021

CROSSREFS

Cf. A038548, A048272, A113652, A228441, A305152, A333781, A348660, A348951, A348952.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Nov 04 2021

STATUS

approved

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