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.

Previous Showing 11-20 of 36 results. Next

A299119 Positive solution to 2^(n-1) = (1/n) * Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 2, 6, 14, 40, 84, 224, 484, 1134, 2480, 5632, 12036, 26624, 56896, 122640, 261078, 557056, 1176876, 2490368, 5237360, 11008704, 23057408, 48234496, 100635144, 209714400, 436154368, 905962860, 1878931264, 3892314112, 8052800160, 16642998272, 34359209436
Offset: 1

Views

Author

Gus Wiseman, Feb 03 2018

Keywords

Comments

For prime p, a(p) = 2^(p-2)*p. - Jon E. Schoenfield, Feb 03 2018

Links

Crossrefs

Cf. A000005, A000010, A000740, A001787, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299149, A299151.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1, n*2^(n-2)-
       add(a(d)*a(n/d), d=divisors(n) minus {1, n})/2)
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Mar 07 2018
  • Mathematica
    nn=50;
sys=Table[2^(n-1)*n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]

A342539 a(n) = Sum_{k=1..n} phi(gcd(k, n))^n.

Original entry on oeis.org

1, 2, 10, 19, 1028, 132, 279942, 65798, 10078726, 2097160, 100000000010, 16797702, 106993205379084, 156728328204, 35186519703560, 281479271809036, 295147905179352825872, 203119914385420, 708235345355337676357650, 1152924803145924620, 46005163783270994804748, 20000000000000000000020
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A332517, A342487, A342534, A342535, A342540, A342541, A342543.

Programs

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

Formula

a(n) = Sum_{d|n} phi(n/d) * phi(d)^n.
If p is prime, a(p) = p-1 + (p-1)^p.
a(n) = Sum_{k=1..n} phi(n/gcd(n,k))^(n-1)*phi(gcd(n,k)). - Richard L. Ollerton, May 09 2021

A342540 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n-1).

Original entry on oeis.org

1, 2, 6, 11, 260, 68, 46662, 16518, 1680134, 524296, 10000000010, 4204550, 8916100448268, 26121388044, 4398583447560, 35185445896204, 18446744073709551632, 33853319413772, 39346408075296537575442, 144116012711673868, 3833767304764361539596, 2000000000000000000020
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A342471, A342539, A342542, A342544.

Programs

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

Formula

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

A342541 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n)).

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 12, 14, 28, 28, 20, 62, 24, 54, 272, 68, 32, 198, 36, 676, 1224, 130, 44, 1348, 4136, 180, 3540, 3426, 56, 12632, 60, 1640, 22520, 304, 129456, 22370, 72, 378, 101808, 270952, 80, 192996, 84, 40630, 1867184, 550, 92, 551528, 1679700, 4198860, 2105408
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A342485, A342539, A342542, A342543.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^(n/#) &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(gcd(k, n))^(n/gcd(k, n)));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^(n/d));

Formula

a(n) = Sum_{d|n} phi(n/d) * phi(d)^(n/d).
If p is prime, a(p) = 2 *(p-1).

A342542 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 9, 15, 13, 11, 33, 13, 19, 105, 33, 17, 91, 19, 209, 469, 31, 23, 641, 1045, 37, 1627, 841, 29, 4217, 31, 673, 10461, 49, 29785, 10281, 37, 55, 49465, 68769, 41, 65197, 43, 12281, 529625, 67, 47, 273185, 279979, 1049661, 1049121, 52657, 53, 803647
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A342473, A342540, A342541, A342544.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^(n/#-1) &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)
  • PARI
    a(n) = sum(k=1, n, eulerphi(gcd(k, n))^(n/gcd(k, n)-1));
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^(n/d-1));

Formula

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

A342544 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(gcd(k, n) - 1).

Original entry on oeis.org

1, 2, 6, 11, 260, 40, 46662, 16398, 1679630, 262408, 10000000010, 4194366, 8916100448268, 13060740684, 4398046511640, 35184372105244, 18446744073709551632, 16926661124436, 39346408075296537575442, 144115188076118572, 3833759992447475215524, 1000000000010000000020
Offset: 1

Views

Author

Seiichi Manyama, Mar 15 2021

Keywords

Links

Crossrefs

Cf. A000010, A029935, A309369, A342436, A342540, A342542, A342543.

Programs

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

Formula

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

A347134 a(n) = Sum_{d|n} phi(n/d) * A069359(d), where phi is Euler totient function.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 8, 5, 12, 1, 23, 1, 16, 14, 20, 1, 36, 1, 35, 18, 24, 1, 60, 9, 28, 21, 47, 1, 87, 1, 48, 26, 36, 22, 103, 1, 40, 30, 92, 1, 119, 1, 71, 66, 48, 1, 148, 13, 92, 38, 83, 1, 144, 30, 124, 42, 60, 1, 247, 1, 64, 86, 112, 34, 183, 1, 107, 50, 183, 1, 268, 1, 76, 110, 119, 34, 215, 1, 228, 81, 84
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2021

Keywords

Comments

Dirichlet convolution of A000010 (Euler totient function phi) with A069359.
Dirichlet convolution of A001221 (omega) with A029935 (the convolution square of Euler phi).

Links

Crossrefs

Cf. A000010, A001221, A069359, A029935.
Cf. also A347131, A347133, A347135.

Programs

Formula

a(n) = Sum_{d|n} A000010(n/d) * A069359(d)
a(n) = Sum_{d|n} A001221(n/d) * A029935(d).
a(n) = Sum_{k=1..n} A069359(gcd(n,k)). - Antti Karttunen, Oct 17 2021

A029936 Number of cusps of Gamma_1(n)\P_1(Q).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 16, 14, 16, 16, 18, 20, 24, 20, 22, 24, 28, 24, 30, 30, 28, 32, 30, 32, 40, 32, 48, 40, 36, 36, 48, 48, 40, 48, 42, 50, 64, 44, 46, 56, 60, 56, 64, 60, 52, 60, 80, 72, 72, 56
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 158.

Links

Programs

  • Mathematica
    a[1] = 1; a[2] = 2; a[4] = 3; a[n_] := DivisorSum[n, EulerPhi[#]* EulerPhi[n/#]&]/2; Array[a, 60] (* Jean-François Alcover, Oct 03 2016 *)
  • PARI
    for(n=1,30, print1(if(n==1, 1, if(n==2, 2, if(n==3, 2, if(n==4, 3, sumdiv(n, d, eulerphi(d)*eulerphi(n/d))/2)))), ", ")) \\ G. C. Greubel, Dec 13 2017

Formula

Except for n=1, 2, 4, this is A029935(n)/2.
a(n) = (1/2)*Sum_{d divides n} phi(d)*phi(n/d), with a(1)=1, a(2)=2, a(3)=2, a(4)=3, and phi(n) = A000010(n). - G. C. Greubel, Dec 13 2017

A338164 Dirichlet g.f.: (zeta(s-2) / zeta(s))^2.

Original entry on oeis.org

1, 6, 16, 33, 48, 96, 96, 168, 208, 288, 240, 528, 336, 576, 768, 816, 576, 1248, 720, 1584, 1536, 1440, 1056, 2688, 1776, 2016, 2448, 3168, 1680, 4608, 1920, 3840, 3840, 3456, 4608, 6864, 2736, 4320, 5376, 8064, 3360, 9216, 3696, 7920, 9984, 6336, 4416, 13056, 7008, 10656
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 14 2020

Keywords

Comments

Dirichlet convolution of Jordan function J_2 (A007434) with itself.

Links

Crossrefs

Cf. A000005, A007427, A007434, A029935, A034714, A306379, A321322, A338165.

Programs

  • Mathematica
    Jordan2[n_] := Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan2[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 50}]
a[1] = 1; f[p_, e_] := p^(2 e - 4) (p^4 + e (p^2 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 50}]

Formula

Multiplicative with a(p^e) = p^(2*e - 4) * (p^4 + e * (p^2 - 1)^2 - 1).
a(n) = Sum_{d|n} J_2(d) * J_2(n/d).
a(n) = Sum_{d|n} d^2 * tau(d) * A007427(n/d), where tau = A000005.
a(n) = Sum_{d|n} d^2 * A321322(n/d).
(1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^2.
Sum_{k=1..n} a(k) ~ ((3*log(n) + 6*gamma - 1)/(9*zeta(3)^2) - 2*zeta'(3) / (3*zeta(3)^3)) * n^3, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 14 2020
Previous Showing 11-20 of 36 results. Next

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

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