A372633 -id:A372633 - cp's OEIS frontend

A372636 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(j*k) / phi(k).

1, 5, 14, 31, 58, 93, 148, 219, 306, 407, 550, 695, 898, 1103, 1323, 1610, 1963, 2293, 2738, 3152, 3597, 4116, 4773, 5362, 6073, 6808, 7611, 8437, 9492, 10348, 11557, 12728, 13868, 15143, 16425, 17753, 19482, 21083, 22687, 24350, 26481, 28186, 30535, 32641

Offset: 1

Seiichi Manyama, May 08 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^3 for n = 1..10000
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A330596, A372619, A372633.

Table[Sum[Sum[EulerPhi[j*k], {j, 1, n}] / EulerPhi[k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, May 08 2024 *)
s = 1; Join[{1}, Table[s += Sum[EulerPhi[j*n] / EulerPhi[j], {j, 1, n}] + Sum[EulerPhi[j*n], {j, 1, n-1}] / EulerPhi[n], {n, 2, 50}]] (* Vaclav Kotesovec, May 08 2024 *)

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

a(n) ~ c * n^3, where c = A330596 / 2 = 0.374267629841... . - Amiram Eldar, May 09 2024

A372661 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} phi(ijk).

Original entry on oeis.org

1, 14, 98, 412, 1436, 3212, 8312, 17460, 34182, 57406, 107306, 161942, 277550, 406490, 581210, 850162, 1292018, 1701752, 2481476, 3216100, 4184464, 5406704, 7414512, 9119640, 11849180, 14736284, 18541664, 22500200, 28950168, 33410736, 42380976, 51166240, 60859420

Offset: 1

Seiichi Manyama, May 09 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Plot of a(n)/n^6 for n = 1..1000
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A372633, A372662, A372663.

Table[Sum[EulerPhi[i*j*k], {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 40}] (* Vaclav Kotesovec, May 09 2024 *)
s = 1; Join[{1}, Table[s += Sum[EulerPhi[i*j*n], {i, 1, n}, {j, 1, n}] + Sum[EulerPhi[i*j*n], {i, 1, n - 1}, {j, 1, n}] + Sum[EulerPhi[i*j*n], {i, 1, n - 1}, {j, 1, n - 1}], {n, 2, 40}]] (* Vaclav Kotesovec, May 09 2024 *)

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

A372674 a(n) = Sum_{j=1..n} Sum_{k=1..n} tau(j*k).

Original entry on oeis.org

1, 8, 23, 54, 89, 162, 221, 326, 439, 596, 707, 964, 1107, 1352, 1645, 1976, 2179, 2630, 2865, 3390, 3859, 4316, 4615, 5406, 5883, 6444, 7059, 7892, 8299, 9430, 9877, 10794, 11635, 12424, 13361, 14852, 15415, 16324, 17349, 18952, 19587, 21342, 22017, 23486, 25177

Offset: 1

Vaclav Kotesovec, May 10 2024

nonn

For m>=1, Sum_{j=1..n} tau(m*j) = A018804(m) * n * log(n) + O(n).

If p is prime, then Sum_{j=1..n} tau(p*j) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n) / (n^2 * (log(n) + 2*gamma - 1/2)^2) for n = 1..100000

Cf. A372633, A372675.
Cf. A006218, A263086.
cf. A000005, A099777, A372713.

Table[Sum[DivisorSigma[0, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[0, n^2] + 2*Sum[DivisorSigma[0, j*n], {j, 1, n - 1}], {n, 2, 50}]]

A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

Original entry on oeis.org

1, 14, 59, 190, 401, 914, 1499, 2632, 4113, 6424, 8645, 13284, 17023, 23092, 30715, 40484, 48711, 63890, 75351, 95792, 116421, 139822, 159911, 199176, 229499, 267438, 309283, 364462, 404933, 482792, 532553, 611208, 688593, 772540, 862471, 998760, 1083615, 1200328

Offset: 1

Vaclav Kotesovec, May 10 2024

nonn

Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^4 for n = 1..100000

Cf. A069097, A372633, A372674.
Cf. A024916, A326124.
Cf. A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.

Table[Sum[DivisorSigma[1, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[1, n^2] + 2*Sum[DivisorSigma[1, j*n], {j, 1, n - 1}], {n, 2, 50}]]

a(n) ~ c * n^4, where c = Pi^4 / (144*zeta(3)) = 0.56274...

A372635 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(j)*phi(k).

Original entry on oeis.org

1, 4, 16, 36, 100, 144, 324, 484, 784, 1024, 1764, 2116, 3364, 4096, 5184, 6400, 9216, 10404, 14400, 16384, 19600, 22500, 29584, 32400, 40000, 44944, 52900, 58564, 72900, 77284, 94864, 104976, 118336, 129600, 147456, 156816, 186624, 202500, 224676, 240100, 280900

Offset: 1

Seiichi Manyama, May 08 2024

nonn

Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A002088, A372633.

nmax = 50; Accumulate[Table[EulerPhi[j], {j, 1, nmax}]]^2 (* Vaclav Kotesovec, May 08 2024 *)

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

PARI
```
a(n) = sum(k=1, n, eulerphi(k))^2;
```

a(n) = A002088(n)^2.

a(n) ~ 9*n^4/(Pi^4). - Vaclav Kotesovec, May 08 2024

cp's OEIS Frontend

A372636 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(j*k) / phi(k).

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A372661 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} phi(i*j*k).

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A372674 a(n) = Sum_{j=1..n} Sum_{k=1..n} tau(j*k).

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A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

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Formula

A372635 a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(j)*phi(k).

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Mathematica

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PARI

Formula

A372661 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} phi(ijk).