A379718 -id:A379718 - cp's OEIS frontend

A379717 The second Jordan totient function applied to the cubefree numbers.

1, 3, 8, 12, 24, 24, 48, 72, 72, 120, 96, 168, 144, 192, 288, 216, 360, 288, 384, 360, 528, 600, 504, 576, 840, 576, 960, 960, 864, 1152, 864, 1368, 1080, 1344, 1680, 1152, 1848, 1440, 1728, 1584, 2208, 2352, 1800, 2304, 2016, 2808, 2880, 2880, 2520, 3480, 2304

Offset: 1

Amiram Eldar, Dec 31 2024

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

Cf. A002117, A004709, A007434, A013661, A358039 (analogous with J_1 = phi), A379715, A379716, A379718.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := Max[FactorInteger[n][[;;, 2]]] < 3; j2 /@ Select[Range[100], cubeFreeQ]

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
iscubefree(n) = if(n == 1, 1, vecmax(factor(n)[, 2]) < 3);
list(lim) = apply(j2, select(iscubefree, vector(lim, i, i)));

a(n) = A007434(A004709(n)).
Sum_{n>=1} 1/a(n) = zeta(2) * zeta(4) / zeta(8) = 35 / (2*Pi^2) = 1.77312071374091100026... .
In general, Sum_{m cubefree} 1/J_k(m) = zeta(k) * zeta(2*k) / zeta(4*k), for k >= 2, where J_k is the k-th Jordan totient function.
In general, Sum_{m k-free} 1/J_2(m) = zeta(2)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^(2*k)), for k >= 2.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3)^3 * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 1.23061243656940899916... . - Amiram Eldar, Jan 03 2025

A379715 The second Jordan totient function applied to the squarefree numbers.

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 504, 840, 576, 960, 960, 864, 1152, 1368, 1080, 1344, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 2880, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224, 3456, 5040, 5328, 4104, 5760, 4032

Offset: 1

Amiram Eldar, Dec 30 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mohammadreza Esfandiari, On the Means of Jordan's Totient Function, Bull. Iran. Math. Soc., Vol. 46 (2020), pp. 1753-1765.
R. Sitaramachandrarao, On an error term of Landau - II, Rocky Mountain J. Math., Vol. 15, No. 2 (1985), pp. 579-588. See p. 581.

Cf. A005117, A007434, A013661, A049200 (analogous with J_1 = phi), A330523, A379716, A379717, A379718.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; j2 /@ Select[Range[100], SquareFreeQ]

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
list(lim) = apply(j2, select(issquarefree, vector(lim, i, i)));

a(n) = A007434(A005117(n)).
Sum_{n>=1} 1/a(n) = zeta(2) (A013661) (Sitaramachandrarao, 1985).
In general, Sum_{m squarefree} 1/J_k(m) = zeta(k), for k >= 2, where J_k is the k-th Jordan totient function.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2)^3 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) =  A013661^3 * A330523 = 2.38520727393117206135... . - Amiram Eldar, Jan 03 2025

A379716 The second Jordan totient function applied to the powerful numbers: a(n) = A007434(A001694(n)).

Original entry on oeis.org

1, 12, 48, 72, 192, 600, 648, 768, 864, 2352, 3072, 3456, 5832, 7200, 7776, 14520, 15000, 12288, 13824, 28392, 28224, 28800, 31104, 43200, 52488, 49152, 55296, 83232, 69984, 115248, 129960, 112896, 115200, 124416, 169344, 174240, 180000, 196608, 279312, 221184, 375000

Offset: 1

Amiram Eldar, Dec 30 2024

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

Cf. A001694, A007434, A323333 (analogous with J_1 = phi), A379715, A379717, A379718.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; seq[lim_] := j2 /@ Union[Flatten[Table[i^2*j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}]]]; seq[1000]

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
list(lim) = apply(j2, select(ispowerful, vector(lim, i, i)));

Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^4) = 1.13107206648894940601... .

In general, Sum_{m powerful} 1/J_k(m) = zeta(k)^2 * Product_{p prime} (1 - 2/p^k + 2/p^(2*k)), for k >= 2, where J_k is the k-th Jordan totient function.

A379832 The second Jordan totient function applied to the exponentially odd numbers.

Original entry on oeis.org

1, 3, 8, 24, 24, 48, 48, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 384, 504, 648, 840, 576, 960, 768, 960, 864, 1152, 1368, 1080, 1344, 1152, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 1944, 2880, 2304, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224

Offset: 1

Amiram Eldar, Jan 03 2025

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

Cf. A007434, A065463, A185197, A268335, A374456 (analogous with J_1 = phi), A379715, A379716, A379717, A379718, A379833.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; expoddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; j2 /@ Select[Range[100], expoddQ]

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(j2, select(isexpodd, vector(lim, i, i)));

a(n) = A007434(A268335(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 1/(p*(p+1)))^3) = A185197 / A065463^3 = 0.57968779180803379088... .
Sum_{n>=1} 1/a(n) = (Pi^6/540) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.67479534964539923068...
In general, Sum_{m exponentially odd} 1/J_k(m) = zeta(k) * zeta(2*k) * Product_{p prime} (1 - 1/p^(2*k) + 1/p^(3*k)), for k >= 2, where J_k is the k-th Jordan totient function.

A379833 The second Jordan totient function applied to the squares.

Original entry on oeis.org

1, 12, 72, 192, 600, 864, 2352, 3072, 5832, 7200, 14520, 13824, 28392, 28224, 43200, 49152, 83232, 69984, 129960, 115200, 169344, 174240, 279312, 221184, 375000, 340704, 472392, 451584, 706440, 518400, 922560, 786432, 1045440, 998784, 1411200, 1119744, 1872792

Offset: 1

Amiram Eldar, Jan 03 2025

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

Cf. A000290, A002117, A002618 (analogous with J_1 = phi), A007434, A379715, A379716, A379717, A379718, A379832.

a:= n-> mul((i[1]^2-1)*i[1]^(4*i[2]-2), i=ifactors(n)[2]):
seq(a(n), n=1..37);  # Alois P. Heinz, Jan 03 2025

f[p_, e_] := (p^2 - 1) * p^(4*e - 2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(4*f[i,2] - 2));}

a(n) = J_2(n^2) = A007434(A000290(n)).
Multiplicative with a(p^e) = (p^2-1) * p^(4*e-2).
Dirichlet g.f.: zeta(s-4)/zeta(s-2).
In general, Dirichlet g.f. of J_k(n^m): zeta(s-m*k)/zeta(s-m*k+k), where J_k is the k-th Jordan totient function.
Sum_{i=1..n} a(i) ~ n^5 / (5*zeta(3)).
In general, Sum_{i=1..n} J_k(i^m) ~ n^(k*m+1) / ((k*m+1)*zeta(k+1)) for k,m >= 1.
Sum_{n>=1} 1/a(n) = (Pi^6/540) * Product_{p prime} (1 - 1/p^2 + 1/p^6) = 1.10666099915727116962...
In general, Sum_{n>=1} 1/J_k(n^m) = zeta(k) * zeta(k*m) * Product_{p prime} (1 - 1/p^k + 1/p^(k*m+k)), for k,m >= 2, and zeta(2) * zeta(m) * Product_{p prime} (1 - 1/p^2 + 1/p^(m+1) + 1/p^(m+2)) for k = 1 and m >= 2.

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A379717 The second Jordan totient function applied to the cubefree numbers.

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A379715 The second Jordan totient function applied to the squarefree numbers.

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A379716 The second Jordan totient function applied to the powerful numbers: a(n) = A007434(A001694(n)).

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A379832 The second Jordan totient function applied to the exponentially odd numbers.

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A379833 The second Jordan totient function applied to the squares.

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