A046970 -id:A046970 - cp's OEIS frontend

A069891 a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.

0, 1, 3, 6, 7, 12, 18, 25, 27, 28, 38, 49, 52, 65, 79, 94, 95, 112, 114, 133, 138, 159, 181, 204, 210, 211, 237, 240, 247, 276, 306, 337, 339, 372, 406, 441, 442, 479, 517, 556, 566, 607, 649, 692, 703, 708, 754, 801, 804, 805, 807, 858, 871, 924, 930, 985, 999

Offset: 0

Dean Hickerson, Apr 09 2002

nonn

Sum_{k=1..n, k squarefree} (1/k) = Sum{k=1..n} (mu(k)^2/k) = (1/zeta(2))*(log(n) + gamma - 2*zeta'(2)/zeta(2)) + O(1/sqrt(n)). (Suryanarayana)

D. Suryanarayana, Asymptotic formula for sum_{n <= x} mu^{2}(n)/n, Indian J. Math. 9 (1967/1968) 543-545.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, 2017, no. 7, pp. 331-338.

Cf. A007913, A046970, A069087.

[0] cat [&+[Squarefree(k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Dec 19 2019

a[n_] := Sum[If[d == 1, 1, Times@@(1-#1[[1]]^2&) /@ FactorInteger[d]] * Binomial[Floor[n/d^2]+1, 2], {d, 1, Floor[Sqrt[n]]}]; Array[a, 100, 0] (* corrected by Amiram Eldar, Apr 02 2020 *)

a(n) = sum(k=1, n, core(k)); \\ Michel Marcus, Dec 19 2019

a(n) = Sum_{d=1..floor(sqrt(n))} f(d)*binomial(floor(n/d^2)+1, 2) where f(d)=A046970(d) is the product of 1-p^2 over all prime divisors p of d.

a(n) is asymptotic to r*n^2, where r = Pi^2/30 = 0.3289868...

A328639 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-2)).

Original entry on oeis.org

1, -5, -10, 5, -26, 50, -50, -5, 10, 130, -122, -50, -170, 250, 260, 5, -290, -50, -362, -130, 500, 610, -530, 50, 26, 850, -10, -250, -842, -1300, -962, -5, 1220, 1450, 1300, 50, -1370, 1810, 1700, 130, -1682, -2500, -1850, -610, -260, 2650, -2210, -50, 50, -130

Offset: 1

Ilya Gutkovskiy, Oct 23 2019

Dirichlet inverse of A065958.

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

Cf. A008683, A008836, A026424 (positions of negative terms), A046970, A065958, A323363, A328640.

a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^2, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 50}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^2 &], {n, 1, 50}]
f[p_, e_] := (-1)^e*(p^2 + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^2)} \\ Andrew Howroyd, Oct 25 2019

a(1) = 1; a(n) = -Sum_{d|n, dA065958(n/d) * a(d).
a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^2, where lambda = A008836 and mu = A008683.
Multiplicative with a(p^e) = (-1)^e*(p^2 + 1). - Amiram Eldar, Nov 30 2020

A189923 Jordan function J_{-5}(n) multiplied by n^5.

Original entry on oeis.org

1, -31, -242, -31, -3124, 7502, -16806, -31, -242, 96844, -161050, 7502, -371292, 520986, 756008, -31, -1419856, 7502, -2476098, 96844, 4067052, 4992550, -6436342, 7502, -3124, 11510052, -242, 520986, -20511148, -23436248

Offset: 1

Wolfdieter Lang, Jun 16 2011

For the Jordan function J_k see the Comtet and Apostol references.

			a(2) = a(4) = a(8) = ... = 1 - 2^5 = -31.
a(4) = mu(1)*1^5 + mu(2)*2^5 + mu(4)*4^5 = 1 - 32 + 0 = -31.
Sum identity for n=4: a(1)*(4/1)^5 + a(2)*(4/2)^5 + a(4)*(4/4)^5 = 1024 - 31*32 - 31 = 1.

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..200 from Indranil Ghosh)

Cf. A023900, A046970, A063453, A189922, for k=-1..-4.

a[n_] := Sum[ MoebiusMu[d]*d^5, {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Sep 03 2012 *)
f[p_, e_] := (1-p^5); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 08 2020 *)

for(n=1, 200, print1(sumdiv(n, d, moebius(d) * d^5),", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = sumdiv(n, d, moebius(d) * d^5); \\ Michel Marcus, Jan 14 2018

a(n) = J_{-5}(n)*n^5 = Product_{p prime |n} (1-p^5), for n>=2, a(1)=1.
a(n) = Sum_{d|n} mu(d)*d^5 with the Moebius function mu = A008683.
Dirichlet g.f.: zeta(s)/zeta(s-5).
Sum identity: Sum_{d|n} a(n)*(n/d)^5 = 1 for all n>=1.
a(n) = a(rad(n)) with rad(n) = A007947(n), the squarefree kernel of n.
G.f.: Sum_{k>=1} mu(k)*k^5*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017

A307648 G.f. A(x) satisfies: 1/(1 - x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...

Original entry on oeis.org

1, 1, -1, -4, -3, -2, 7, 7, 4, -6, 14, -11, -4, -47, 9, 6, 161, -93, -33, -269, 232, -83, 660, -733, 500, -779, 1527, -2291, 1876, -3892, 5598, -3056, 7791, -14088, 11289, -17113, 28083, -26211, 34645, -60715, 73180, -80951, 111926, -155269, 178561, -233709, 359679, -403884, 454659, -697310, 862133

Offset: 0

Ilya Gutkovskiy, Apr 19 2019

sign

Euler transform of A055615.

			G.f.: A(x) = 1 + x - x^2 - 4*x^3 - 3*x^4 - 2*x^5 + 7*x^6 + 7*x^7 + 4*x^8 - 6*x^9 + 14*x^10 - 11*x^11 - 4*x^12 - 47*x^13 + ...

Cf. A008683, A046970, A055615, A117209, A307649.

terms = 50; CoefficientList[Series[Product[1/(1 - x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 50; CoefficientList[Series[Exp[Sum[Sum[MoebiusMu[d] d^2, {d, Divisors[k]}] x^k/k, {k, 1, terms}]], {x, 0, terms}], x]
terms = 50; A[] = 1; Do[A[x] = 1/((1 - x) Product[A[x^k]^k, {k, 2, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

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

G.f.: exp(Sum_{k>=1} A046970(k)*x^k/k).

A069056 Numbers k such that Sum_{d|k} d^2*mu(d) divides k^2.

Original entry on oeis.org

1, 12, 24, 36, 48, 72, 96, 108, 120, 144, 192, 216, 240, 288, 324, 336, 360, 384, 432, 480, 576, 600, 648, 672, 720, 768, 864, 960, 972, 1008, 1080, 1152, 1200, 1296, 1344, 1440, 1536, 1728, 1800, 1920, 1944, 2016, 2160, 2304, 2352, 2400, 2448, 2592, 2688

Offset: 1

Benoit Cloitre, Apr 07 2002

Numbers k such that A046970(k) divides k^2. [corrected by Amiram Eldar, Apr 20 2025]

If n > 1, a(n+1) - a(n) == 0 (mod 12), so a(n+1) - a(n) = 12 for n=2,3,4,5,7,8,...; a(n+1) - a(n) = 24 for n=6,9,.... Conjecture: if c > 2 and n > 1, Sum_{d|n} d^c*mu(d) never divides n^c. Hence A063453(n) never divides n^3 for n > 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A046970, A063453.

a069056 n = a069056_list !! (n-1)
a069056_list = filter (\x -> x ^ 2 `mod` a046970 x == 0) [1..]
-- Reinhard Zumkeller, Jan 19 2012

f[d_] := d^2*MoebiusMu[d]; ok[n_] := Divisible[n^2, Total[f /@ Divisors[n]]]; Select[Range[3000], ok] (* Jean-François Alcover, Nov 15 2011 *)
q[k_] := Divisible[k^2, Times @@ ((First[#]^2-1)& /@ FactorInteger[k])]; q[1] = True; Select[Range[3000], q] (* Amiram Eldar, Apr 20 2025 *)

for(n=1,1000,if(n^2%sumdiv(n,d,moebius(d)*d^2)==0,print1(n,",")))

isok(k) = {my(f = factor(k)); !(k^2 % prod(i = 1, #f~, f[i,1]^2-1));} \\ Amiram Eldar, Apr 20 2025

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.

A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).

Original entry on oeis.org

1, -5, -11, 3, -29, 55, -55, 3, 16, 145, -131, -33, -181, 275, 319, 3, -305, -80, -379, -87, 605, 655, -551, -33, 96, 905, 16, -165, -869, -1595, -991, 3, 1441, 1525, 1595, 48, -1405, 1895, 1991, -87, -1721, -3025, -1891, -393, -464, 2755, -2255, -33, 288, -480, 3355, -543, -2861, -80, 3799

Offset: 1

Ilya Gutkovskiy, Feb 16 2022

Dirichlet inverse of A069097.

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

Cf. A000005, A023900, A046970, A055615, A069097, A101035, A328254, A334657.

A069097[n_] := Sum[GCD[n, k]^2, {k, 1, n}]; a[1] = 1; a[n_] := a[n] = -Sum[A069097[n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := If[e == 1, 0, p^3] - p^2 - p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA069097(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d));
v351654 = DirInverseCorrect(vector(up_to, n, A069097(n)));
A351654(n) = v351654[n]; \\ Antti Karttunen, Feb 16 2022

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)*(1 - p^2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022

a(1) = 1; a(n) = -Sum_{d|n, d < n} A069097(n/d) * a(d).
a(n) = Sum_{d|n} A023900(n/d) * A334657(d).
a(n) = Sum_{d|n} A046970(n/d) * A055615(d).
a(n) = Sum_{d|n} A000005(n/d) * A328254(d).
Multiplicative with a(p) = -p^2 - p + 1, and a(p^e) = p^3 - p^2 - p + 1 for e >= 2. - Amiram Eldar, May 25 2025

A307656 G.f. A(x) satisfies: (1 - x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...

Original entry on oeis.org

1, -1, 2, 1, 0, 8, -7, 22, -6, 13, 29, -11, 82, -36, 114, 13, 103, 88, 88, 275, -20, 549, -200, 1007, -144, 811, 730, 188, 2093, -777, 3538, -643, 4083, -537, 4562, 2478, 1973, 8062, -3508, 17362, -8164, 20281, -2227, 17483, 8605, 2946, 30190, -6085, 53176, -28913, 78516

Offset: 0

Ilya Gutkovskiy, Apr 20 2019

sign

Convolution inverse of A307648.

			G.f.: A(x) = 1 - x + 2*x^2 + x^3 + 8*x^5 - 7*x^6 + 22 x^7 - 6*x^8 + 13*x^9 + 29*x^10 - 11*x^11 + 82*x^12 - 36*x^13 + ...

Cf. A008683, A046970, A055615, A117208, A307648.

terms = 50; CoefficientList[Series[Product[(1 - x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 50; CoefficientList[Series[Exp[-Sum[Sum[MoebiusMu[d] d^2, {d, Divisors[k]}] x^k/k, {k, 1, terms}]], {x, 0, terms}], x]
terms = 50; A[] = 1; Do[A[x] = (1 - x)/Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

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

G.f.: exp(-Sum_{k>=1} A046970(k)*x^k/k).

cp's OEIS Frontend

A069891 a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.

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A328639 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-2)).

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A189923 Jordan function J_{-5}(n) multiplied by n^5.

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A307648 G.f. A(x) satisfies: 1/(1 - x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

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A069056 Numbers k such that Sum_{d|k} d^2*mu(d) divides k^2.

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

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A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).

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A328639 Dirichlet g.f.: zeta(2s) / (zeta(s) zeta(s-2)).

A307648 G.f. A(x) satisfies: 1/(1 - x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...

A307656 G.f. A(x) satisfies: (1 - x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...