cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.

Original entry on oeis.org

1, 9, 29, 74, 129, 261, 349, 596, 789, 1161, 1341, 2146, 2209, 3141, 3741, 4776, 4929, 7101, 6877, 9546, 10121, 12069, 12189, 17284, 16145, 19881, 21321, 25826, 24417, 33669, 29821, 38224, 38889, 44361, 45021, 58386, 50689, 61893, 64061, 76884, 68961, 91089, 79549, 99234, 101781
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Links

Crossrefs

Column 3 of A343510.
Cf. A000010, A001157 (sigma_2(n)), A018804, A054610, A069097, A309323, A332517, A342423, A342433, A343498, A343499, A343513.

Programs

  • Magma
    A343497:= func< n | (&+[d^3*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343497(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024
  • Maple
    with(numtheory):
seq(add(phi(n/d) * d^3, d in divisors(n)), n = 1..50); # Peter Bala, Jan 20 2024
  • Mathematica
    a[n_] := Sum[GCD[k, n]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e - 1)*((p^2 + p + 1)*p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
A343497[n_]:= DivisorSum[n, #^3*EulerPhi[n/#] &]; Table[A343497[n], {n, 50}] (* G. C. Greubel, Jun 24 2024 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^3);
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^3);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 2));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4))
  • SageMath
    def A343497(n): return sum(k^3*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343497(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

Formula

a(n) = Sum_{d|n} phi(n/d) * d^3.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_2(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
Dirichlet g.f.: zeta(s-1) * zeta(s-3) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 45*zeta(3)*n^4 / (2*Pi^4). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*((p^2+p+1)*p^(2*e) - 1)/(p+1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n) = Sum_{d divides n} d * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 20 2024

A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.

Original entry on oeis.org

1, 17, 83, 274, 629, 1411, 2407, 4388, 6729, 10693, 14651, 22742, 28573, 40919, 52207, 70216, 83537, 114393, 130339, 172346, 199781, 249067, 279863, 364204, 393145, 485741, 545067, 659518, 707309, 887519, 923551, 1123472, 1216033, 1420129, 1514003, 1843746, 1874197
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Links

Crossrefs

Column 4 of A343510.
Cf. A000010, A001158 (sigma_3(n)), A018804, A054611, A069097, A332517, A342423, A342433, A343497, A343499, A343514.

Programs

  • Magma
    A343498:= func< n | (&+[d^4*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343498(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024
  • Mathematica
    a[n_] := Sum[GCD[k, n]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^4);
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^4);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 3));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^5))
  • SageMath
    def A343498(n): return sum(k^4*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343498(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

Formula

a(n) = Sum_{d|n} phi(n/d) * d^4.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_3(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5.
Dirichlet g.f.: zeta(s-1) * zeta(s-4) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (450*zeta(5)). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i, j, k, l, n) = Sum_{d divides n} d * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 18 2024

A343499 a(n) = Sum_{k=1..n} gcd(k, n)^5.

Original entry on oeis.org

1, 33, 245, 1058, 3129, 8085, 16813, 33860, 59541, 103257, 161061, 259210, 371305, 554829, 766605, 1083528, 1419873, 1964853, 2476117, 3310482, 4119185, 5315013, 6436365, 8295700, 9778145, 12253065, 14468481, 17788154, 20511177, 25297965, 28629181, 34672912, 39459945, 46855809
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Links

Crossrefs

Column 5 of A343510.
Cf. A000010, A001159 (sigma_4(n)), A018804, A054612, A059378, A069097, A332517, A342423, A342433, A343497, A343498, A344524.

Programs

  • Magma
    A343499:= func< n | (&+[d^5*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343499(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024
  • Mathematica
    a[n_] := Sum[GCD[k, n]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^5);
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^5);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 4));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^6))
  • SageMath
    def A343499(n): return sum(k^5*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343499(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

Formula

a(n) = Sum_{d|n} phi(n/d) * d^5.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_4(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6.
Dirichlet g.f.: zeta(s-1) * zeta(s-5) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 315*zeta(5)*n^6 / (2*Pi^6). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_5 <= n} gcd(i_1, ..., i_5, n) = Sum_{d divides n} d * J_5(n/d), where the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 29 2024

A342424 a(n) = Sum_{k=1..n} gcd(k,n)^(n/gcd(k,n)).

Original entry on oeis.org

1, 3, 5, 10, 9, 33, 13, 60, 69, 167, 21, 470, 25, 837, 1245, 1624, 33, 5067, 37, 9570, 13841, 20633, 45, 54612, 12545, 98511, 119601, 201646, 57, 562957, 61, 794928, 1774185, 2097491, 536037, 5381754, 73, 9437601, 19136333, 14296940
Offset: 1

Views

Author

Seiichi Manyama, Mar 11 2021

Keywords

Links

Crossrefs

Cf. A000010, A332517, A342423.

Programs

  • Mathematica
    a[n_] := Sum[GCD[k, n]^(n/GCD[k, n]), {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Mar 11 2021 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^(n/gcd(k, n)));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n/d));

Formula

a(n) = Sum_{d|n} phi(n/d) * d^(n/d).

A342436 a(n) = Sum_{k=1..n} gcd(k,n)^(gcd(k,n) - 1).

Original entry on oeis.org

1, 3, 11, 68, 629, 7791, 117655, 2097224, 43046745, 1000000637, 25937424611, 743008378618, 23298085122493, 793714773371811, 29192926025391919, 1152921504608944272, 48661191875666868497, 2185911559738739594277, 104127350297911241532859
Offset: 1

Views

Author

Seiichi Manyama, Mar 12 2021

Keywords

Links

Crossrefs

Cf. A000010, A056665, A342421, A342423, A342435.

Programs

  • Mathematica
    a[n_] := Sum[GCD[k, n]^(GCD[k, n] - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^(gcd(k, n)-1));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^(d-1));

Formula

a(n) = Sum_{d|n} phi(n/d) * d^(d-1).

A342449 a(n) = Sum_{k=1..n} gcd(k,n)^k.

Original entry on oeis.org

1, 5, 29, 262, 3129, 46705, 823549, 16777544, 387421251, 10000003469, 285311670621, 8916100581446, 302875106592265, 11112006826387025, 437893890391180013, 18446744073743123788, 827240261886336764193, 39346408075299116257065
Offset: 1

Views

Author

Seiichi Manyama, Mar 13 2021

Keywords

Crossrefs

Cf. A000010, A001923, A018804, A050873, A231712, A332517, A341036, A342370, A342389, A342423.

Programs

  • Mathematica
    a[n_] := Sum[GCD[k, n]^k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^k);

Formula

If p is prime, a(p) = p-1 + p^p = A231712(p).

A342543 a(n) = Sum_{k=1..n} phi(gcd(k, n))^gcd(k, n).

Original entry on oeis.org

1, 2, 10, 19, 1028, 76, 279942, 65558, 10077718, 1049608, 100000000010, 16777334, 106993205379084, 78364444044, 35184372090920, 281474976776236, 295147905179352825872, 101559966746268, 708235345355337676357650, 1152921504607897676, 46005119909369702026044, 10000000000100000000020
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A342423, A342488, A342539, A342541, A342544.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^# &]; Array[a, 20] (* Amiram Eldar, Mar 15 2021 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(gcd(k, n))^gcd(k, n));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^d);

Formula

a(n) = Sum_{d|n} phi(n/d) * phi(d)^d.
If p is prime, a(p) = p-1 + (p-1)^p.

A342435 a(n) = Sum_{k=1..n} gcd(k,n)^(gcd(k,n) - 2).

Original entry on oeis.org

1, 2, 5, 19, 129, 1303, 16813, 262166, 4782981, 100000133, 2357947701, 61917365564, 1792160394049, 56693912392115, 1946195068359645, 72057594038190124, 2862423051509815809, 121439531096599037355
Offset: 1

Views

Author

Seiichi Manyama, Mar 12 2021

Keywords

Crossrefs

Cf. A000010, A342423, A342436.

Programs

  • Mathematica
    a[n_] := Sum[GCD[k, n]^(GCD[k, n] - 2), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^(gcd(k, n)-2));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^(d-2));

Formula

a(n) = Sum_{d|n} phi(n/d) * d^(d-2).

A344194 a(n) = Sum_{k=1..n} tau(gcd(k,n))^gcd(k,n), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 10, 87, 36, 4114, 134, 65629, 19705, 1048628, 2058, 2176786622, 8204, 268435614, 1073741928, 152587956347, 131088, 101559956696337, 524306, 3656158441111964, 4398046511420, 17592186046514, 8388630, 4722366482871822135514, 847288609591, 4503599627378748
Offset: 1

Views

Author

Seiichi Manyama, May 11 2021

Keywords

Crossrefs

Cf. A000005, A342423, A344081, A344140, A344195.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^# &]; Array[a, 26] (* Amiram Eldar, May 11 2021 *)
  • PARI
    a(n) = sum(k=1, n, numdiv(gcd(k, n))^gcd(k, n));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^d);

Formula

a(n) = Sum_{d|n} phi(n/d) * tau(d)^d.
If p is prime, a(p) = 2^p - 1 + p.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.