A372636 -id:A372636 - cp's OEIS frontend

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

7, 4, 8, 5, 3, 5, 2, 5, 9, 6, 8, 2, 3, 6, 3, 5, 6, 4, 6, 4, 4, 2, 1, 5, 0, 4, 8, 6, 3, 7, 9, 1, 0, 6, 0, 1, 6, 4, 1, 6, 4, 0, 3, 4, 3, 0, 0, 5, 3, 2, 4, 4, 0, 4, 5, 1, 5, 8, 5, 2, 7, 9, 3, 9, 2, 5, 9, 2, 5, 5, 8, 6, 8, 9, 5, 4, 9, 5, 8, 8, 3, 4, 2, 1, 2, 6, 2, 0, 6, 8, 1, 4, 6, 4, 7, 0, 9, 8, 1, 3, 1, 4, 3, 3, 5, 4

Offset: 0

Vaclav Kotesovec, Dec 19 2019

The asymptotic density of A337050. - Amiram Eldar, Aug 13 2020

			0.748535259682363564644215048637910601641640343005324404515852793925925...

Cf. A057723, A065464, A065473, A065476, A065487, A328017, A330594, A330595, A337050, A372636.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 - 1/p^2 + 1/p^3) \\ Amiram Eldar, Mar 17 2021

Equals (6/Pi^2) * A065487. - Amiram Eldar, Jun 10 2020

A372662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} phi(ijk) / phi(k).

Original entry on oeis.org

1, 14, 72, 271, 731, 1649, 3417, 6654, 11495, 18635, 29971, 45088, 67562, 95862, 129315, 180548, 247178, 324463, 429785, 547025, 679318, 853788, 1080248, 1324791, 1617620, 1964622, 2361036, 2813206, 3392778, 3946537, 4704245, 5550066, 6389704, 7421750, 8455877

Offset: 1

Seiichi Manyama, May 09 2024

nonn

Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A372636, A372661, A372663.

Table[Sum[EulerPhi[i*j*k]/EulerPhi[k], {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 40}] (* Vaclav Kotesovec, May 10 2024 *)

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

A372664 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(2jk) / phi(k).

Original entry on oeis.org

1, 9, 21, 55, 93, 163, 241, 383, 507, 709, 915, 1205, 1497, 1907, 2220, 2794, 3306, 3966, 4610, 5438, 6075, 7113, 8067, 9245, 10272, 11742, 12900, 14552, 16086, 17798, 19552, 21894, 23532, 26082, 27933, 30589, 33105, 36307, 38619, 41945, 45049, 48459, 51875

Offset: 1

Seiichi Manyama, May 09 2024

nonn

Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A372636, A372665.

Table[Sum[Sum[EulerPhi[2*j*k], {j, 1, n}]/EulerPhi[k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, May 09 2024 *)

a(n) = sum(j=1, n, sum(k=1, n, eulerphi(2*j*k)/eulerphi(k)));

cp's OEIS Frontend

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

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A372662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} phi(ijk) / phi(k).

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A372664 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(2jk) / phi(k).

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cp's OEIS Frontend

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A372662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} phi(i*j*k) / phi(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A372664 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(2*j*k) / phi(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A372662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} phi(ijk) / phi(k).

A372664 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(2jk) / phi(k).