A322360 -id:A322360 - cp's OEIS frontend

A046970 Dirichlet inverse of the Jordan function J_2 (A007434).

1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576

Offset: 1

Douglas Stoll, dougstoll(AT)email.msn.com

B(n+2) = -B(n)*((n+2)*(n+1)/(4*Pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4*Pi^2)) * Sum_{j>=1} a(j)/j^(n+2).

Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002

			a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
a(4) = -3 because the divisors of 4 are {1, 2, 4} and mu(1)*1^2 + mu(2)*2^2 + mu(4)*4^2 = -3.
E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - _Jon Perry_, Aug 24 2010
G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + ...

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
David M. Smith, Multiple-Precision Evaluation of Bernoulli Numbers, Loyola Marymount Univ., 2024. See p. 3.
Wikipedia, Riemann zeta function.

Dirichlet inverse of Jordan totient function J_r(n): A023900 (r = 1), A063453(r = 3), A189922 (r = 4).

Cf. A007434, A027641, A027642, A027748, A046692, A076479, A084920, A322360.

a046970 = product . map ((1 -) . (^ 2)) . a027748_row
-- Reinhard Zumkeller, Jan 19 2012

Jinvk := proc(n,k) local a,f,p ; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; a := a*(1-p^k) ; end do: a ; end proc:
A046970 := proc(n) Jinvk(n,2) ; end proc: # R. J. Mathar, Jul 04 2011

muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez)
Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(1 - x[[i]][[1]]^2)]; p}, {n, 1, 50, 1}]] (* Jon Perry, Aug 24 2010 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ d], {d, Divisors @ n}]]; (* Michael Somos, Jan 11 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (1 - #[[1]]^2 & /@ FactorInteger @ n)]; (* Michael Somos, Jan 11 2014 *)

A046970(n)=sumdiv(n,d,d^2*moebius(d)) \\ Benoit Cloitre

{a(n) = if( n<1, 0, direuler( p=2, n, (1 - X*p^2) / (1 - X))[n])}; /* Michael Somos, Jan 11 2014 */

from math import prod
from sympy import primefactors
def A046970(n): return prod(1-p**2 for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

Multiplicative with a(p^e) = 1 - p^2.
a(n) = Sum_{d|n} mu(d)*d^2.
abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - Jon Perry, Aug 24 2010
From Wolfdieter Lang, Jun 16 2011: (Start)
Dirichlet g.f.: zeta(s)/zeta(s-2).
a(n) = J_{-2}(n)*n^2, with the Jordan function J_k(n), with J_k(1):=1. See the Apostol reference, p. 48. exercise 17. (End)
a(prime(n)) = -A084920(n). - R. J. Mathar, Aug 28 2011
G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017
a(n) = Sum_{d divides n} d * (sigma_1(d))^(-1) * sigma_1(n/d), where (sigma_1(n))^(-1) = A046692(n) denotes the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024
a(n) = A076479(n) * A322360(n). - Amiram Eldar, Feb 02 2024

Corrected and extended by Vladeta Jovovic, Jul 25 2001

Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005

A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 6, 8, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 6, 2, 6, 2, 8, 2, 1, 4, 2, 24, 2, 2, 6, 8, 2, 2, 12, 2, 2, 8, 2, 2, 2, 2, 2, 8, 6, 2, 2, 8, 6, 4, 2, 2, 8, 2, 6, 4, 1, 12, 4, 2, 2, 4, 24, 2, 2, 2, 6, 8, 6, 12, 24, 2, 2, 2, 2, 2, 12, 4, 6, 8, 2, 2, 8, 8, 2, 4, 2, 24, 2, 2, 6, 4

Offset: 1

Labos Elemer, Dec 04 2001

nonn

Frequently equal, but not identical, to A009223 (i.e. GCD of sigma and phi of n).

Antti Karttunen, Table of n, a(n) for n = 1..23374
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A048250, A173557, A023900, A000203, A007947, A000010, A009223, A066087, A322359, A322360, A322362.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Table[g2[w], {w, 1, 128}]
a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[Times@@((#-1)& @@@ f), Times@@((#+1)& @@@ f)]]]; Array[a, 100] (* Amiram Eldar, Dec 05 2018 *)

a(n)=my(f=factor(n)[,1]);gcd(prod(i=1,#f,f[i]+1),prod(i=1,#f,f[i]-1)) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = gcd(A048250(n), A023900(n)) = gcd(A000203(A007947(n)), A000010(A007947(n))).

a(n) = A322360(n) / A322359(n). - Antti Karttunen, Dec 04 2018

Name edited, part of the old name transferred to the formula section by Antti Karttunen, Dec 04 2018

A351265 Sum of the squares of the squarefree divisors of n.

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, 2900, 850, 2810, 50, 3172, 250, 3620

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^2 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(6) = 50; a(6) = Sum_{d|6} d^2 * mu(d)^2 = 1^2*1 + 2^2*1 + 3^2*1 + 6^2*1 = 50.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A253905, A328639, A322360.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), this sequence (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^2); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n],SquareFreeQ]^2],{n,80}] (* Harvey P. Dale, Dec 26 2024 *)

a(n) = sumdiv(n, d, if (issquarefree(d), d^2)); \\ Michel Marcus, Feb 06 2022

a(n) = Sum_{d|n} d^2 * mu(d)^2.
a(n) = abs(A328639(n)).
G.f.: Sum_{k>=1} mu(k)^2 * k^2 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^2. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(3*zeta(2)) = A253905 / 3 = 0.243587... . - Amiram Eldar, Nov 10 2022
Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s-4). - Michael Shamos, Aug 05 2023

A372619 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 2, 5, 6, 1, 3, 5, 9, 10, 1, 2, 5, 7, 13, 12, 1, 3, 4, 9, 11, 17, 18, 1, 2, 6, 6, 13, 14, 23, 22, 1, 3, 4, 10, 11, 17, 20, 31, 28, 1, 2, 5, 6, 14, 13, 23, 24, 37, 32, 1, 3, 5, 9, 10, 20, 19, 31, 33, 45, 42, 1, 2, 5, 7, 13, 12, 26, 23, 37, 37, 55, 46

Offset: 1

Seiichi Manyama, May 07 2024

			Square array T(n,k) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2,  3,  2,  3,  2,  3,  2,  3,  2,  3, ...
   4,  5,  5,  5,  4,  6,  4,  5,  5,  5, ...
   6,  9,  7,  9,  6, 10,  6,  9,  7,  9, ...
  10, 13, 11, 13, 11, 14, 10, 13, 11, 14, ...
  12, 17, 14, 17, 13, 20, 12, 17, 14, 18, ...
  18, 23, 20, 23, 19, 26, 19, 23, 20, 24, ...

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

Columns k=1..10 give: A002088, A049690, A372621, A049690, A372622, A372637, A372638, A049690, A372621, A372639.
Main diagonal gives A070639.
Cf. A000010, A372606.
Cf. A078615, A322360.

T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}] / EulerPhi[k]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2024 *)

T(n, k) = sum(j=1, n, eulerphi(k*j))/eulerphi(k);

T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = A078615(k)/A322360(k) is the multiplicative function defined by c(p^e) = p^2/(p^2-1). - Amiram Eldar, May 09 2024

A322359 Least common multiple of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n.

Original entry on oeis.org

1, 3, 4, 3, 12, 12, 24, 3, 4, 36, 60, 12, 84, 24, 24, 3, 144, 12, 180, 36, 96, 180, 264, 12, 12, 84, 4, 24, 420, 72, 480, 3, 240, 432, 48, 12, 684, 180, 168, 36, 840, 96, 924, 180, 24, 792, 1104, 12, 24, 36, 288, 84, 1404, 12, 360, 24, 720, 1260, 1740, 72, 1860, 480, 96, 3, 336, 720, 2244, 432, 1056, 144, 2520, 12

Offset: 1

Antti Karttunen, Dec 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to lcm's

Cf. A048250, A066086, A173557, A322360.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[Times@@((#-1)& @@@ f), Times@@((#+1)& @@@ f)]]]; Array[a, 100] (* Amiram Eldar, Dec 05 2018 *)

A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322359(n) = lcm(A048250(n), A173557(n));

a(n) = lcm(A048250(n), A173557(n)).

a(n) = A322360(n)/A066086(n).

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54

Offset: 1

José María Grau Ribas, Jan 04 2021

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

			a(2^s) = 3 for all s>0.

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

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021

fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]

A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ Antti Karttunen, Jan 06 2021

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022

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A046970 Dirichlet inverse of the Jordan function J_2 (A007434).

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A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

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A351265 Sum of the squares of the squarefree divisors of n.

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A372619 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).

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A322359 Least common multiple of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n.

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A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

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