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 1-7 of 7 results.

A007557 Shifts left when inverse Moebius transform applied twice.

Original entry on oeis.org

1, 1, 3, 5, 10, 12, 24, 26, 43, 52, 78, 80, 133, 135, 189, 219, 295, 297, 428, 430, 584, 642, 804, 806, 1100, 1123, 1395, 1494, 1856, 1858, 2428, 2430, 2977, 3143, 3739, 3811, 4790, 4792, 5654, 5930, 7072, 7074, 8656
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Equals eigensequence of triangle A127170 (the square of the inverse Mobius transform). - Gary W. Adamson, Apr 27 2009

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000005, A007554, A038045, A038046, A003238, A070965.
Cf. A127170. - Gary W. Adamson, Apr 27 2009

Programs

  • Mathematica
    a[n_] := a[n] = Sum[ DivisorSigma[0, (n - 1)/d]*a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 43}] (* Jean-François Alcover, Dec 12 2011, after Vladeta Jovovic *)

Formula

a(n+1) = Sum_{d divides n} tau(n/d)*a(d). - Vladeta Jovovic, Jan 24 2003
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*x^(i*j)/(1 - x^(i*j))). (End)

Extensions

More terms from Vladeta Jovovic, Jan 24 2003

A332791 a(1) = 1; a(n+1) = Sum_{d|n} phi(d) * a(d).

Original entry on oeis.org

1, 1, 2, 5, 12, 49, 104, 625, 2512, 15077, 60358, 603581, 2414438, 28973257, 173840168, 1390721397, 11125773688, 178012379009, 1068074289230, 19225337206141, 153802697709496, 1845632372514581, 18456323725749392, 406039121966486625, 3248312975734309938
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 24 2020

Keywords

Links

Crossrefs

Cf. A000010, A006874, A038045, A057660, A307793, A307794, A332792.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Sum[EulerPhi[d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 25}]
a[1] = 1; a[n_] := a[n] = Sum[a[(n - 1)/GCD[n - 1, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 25}]

Formula

a(1) = 1; a(n+1) = Sum_{k=1..n} a(n/gcd(n, k)).
a(n) = Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A332792 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(d) * a(d).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 6, 3, 6, 1, 16, 1, 8, 7, 30, 1, 30, 1, 34, 9, 12, 1, 104, 5, 14, 21, 60, 1, 96, 1, 270, 13, 18, 11, 278, 1, 20, 15, 330, 1, 174, 1, 136, 81, 24, 1, 1176, 7, 130, 19, 186, 1, 588, 15, 804, 21, 30, 1, 1204, 1, 32, 135, 4590, 17, 402, 1, 310, 25, 348
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 24 2020

Keywords

Links

Crossrefs

Cf. A000010, A006874, A008578 (positions of 1's), A038045, A057660, A332791.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n, k] > 1, a[n/GCD[n, k]], 0], {k, 1, n}]; Table[a[n], {n, 1, 70}]
  • PARI
    up_to = 20000;
A332792list(n) = { my(v=vector(n)); v[1] = 1; for(n=2, #v, v[n] = sumdiv(n, d, if(d==n, 0, v[d]*eulerphi(d)))); (v); };
v332792 = A332792list(up_to);
A332792(n) = v332792[n]; \\ Antti Karttunen, Jan 22 2025

Formula

a(1) = 1; a(n) = Sum_{k=1..n, gcd(n, k) > 1} a(n/gcd(n, k)).

A338750 a(n) = 1 + Sum_{k=1..n-1} a(gcd(n,k)).

Original entry on oeis.org

1, 2, 3, 5, 5, 10, 7, 14, 13, 18, 11, 35, 13, 26, 31, 41, 17, 58, 19, 65, 45, 42, 23, 122, 41, 50, 63, 95, 29, 154, 31, 122, 73, 66, 83, 241, 37, 74, 87, 230, 41, 226, 43, 155, 193, 90, 47, 419, 85, 194, 115, 185, 53, 338, 135, 338, 129, 114, 59, 679, 61, 122, 283
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 06 2020

Keywords

Comments

Inverse Moebius transform of A006874.

Crossrefs

Cf. A000010, A006874, A038045, A130054.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[GCD[n, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 63}]
a[n_] := a[n] = 1 + DivisorSum[n, EulerPhi[n/#] a[#] &, # < n &]; Table[a[n], {n, 1, 63}]

Formula

G.f. A(x) satisfies: A(x) = x / (1 - x) + Sum_{k>=2} phi(k) * A(x^k).
a(n) = 1 + Sum_{d|n, d < n} phi(n/d) * a(d).
a(n) = Sum_{d|n} A006874(d).

A341639 a(1) = 1; a(n+1) = Sum_{d|n} phi(d) * a(d) * a(n/d).

Original entry on oeis.org

1, 1, 2, 6, 19, 95, 291, 2037, 10203, 71429, 357240, 3929640, 19648533, 255430929, 1788018540, 16092167088, 144829514049, 2462101738833, 17234712244012, 327459532636228, 2947135794083881, 38312765323095109, 421440418557975839, 9693129626833444297, 87238166641520673597
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

Crossrefs

Cf. A000010, A038044, A038045, A332791.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[EulerPhi[d] a[d] a[(n - 1)/d], {d, Divisors[n - 1]}]; Table[a[n], {n, 25}]
a[1] = 1; a[n_] := a[n] = Sum[a[GCD[n - 1, k]] a[(n - 1)/GCD[n - 1, k]], {k, n - 1}]; Table[a[n], {n, 25}]

Formula

a(1) = 1; a(n+1) = Sum_{k=1..n} a(gcd(n,k)) * a(n/gcd(n,k)).

A346114 a(n+1) = 1 + Sum_{k=1..n} a(gcd(n,k)).

Original entry on oeis.org

1, 2, 4, 7, 12, 17, 28, 35, 51, 66, 91, 102, 150, 163, 210, 259, 325, 342, 454, 473, 608, 701, 823, 846, 1099, 1168, 1355, 1500, 1786, 1815, 2290, 2321, 2711, 2954, 3328, 3537, 4302, 4339, 4848, 5221, 6075, 6116, 7269, 7312, 8306, 9059, 9949, 9996, 11795, 12006
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 05 2021

Keywords

Crossrefs

Cf. A000010, A006874, A038045, A338750.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[GCD[n - 1, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 50}]
nmax = 50; A[] = 0; Do[A[x] = x (1/(1 - x) + Sum[EulerPhi[k] A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * (1 / (1 - x) + Sum_{k>=1} phi(k) * A(x^k)).
a(1) = 1; a(n+1) = 1 + Sum_{d|n} phi(n/d) * a(d).

A372618 a(1) = a(2) = 1; a(n+2) = Sum_{k=1..n} a(gcd(n,k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 21, 29, 31, 46, 43, 65, 61, 91, 77, 128, 95, 172, 127, 213, 149, 292, 181, 347, 231, 437, 259, 562, 289, 679, 361, 772, 431, 991, 467, 1104, 565, 1352, 605, 1613, 647, 1877, 835, 2048, 881, 2529, 965, 2802, 1135, 3216, 1187
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Crossrefs

Cf. A000010, A006874, A038045.

Programs

  • Mathematica
    a[1] = a[2] = 1; a[n_] := a[n] = Sum[a[GCD[n - 2, k]], {k, 1, n - 2}]; Table[a[n], {n, 1, 55}]
nmax = 55; A[] = 0; Do[A[x] = x + x^2 (1 + Sum[EulerPhi[k] A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x + x^2 * ( 1 + Sum_{k>=1} phi(k) * A(x^k) ).
a(1) = a(2) = 1; a(n+2) = Sum_{d|n} phi(n/d) * a(d).
Showing 1-7 of 7 results.

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

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