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 1-5 of 5 results.

A360520 a(n) = A120963(n) + A341711(floor(n/2)).

Original entry on oeis.org

2, 3, 11, 15, 43, 57, 137, 177, 389, 495, 1003, 1257, 2421, 3003, 5515, 6771, 12011, 14631, 25143, 30399, 50931, 61197, 100161, 119643, 192051, 228255, 359839, 425631, 660623, 778119, 1190359, 1396479, 2109119, 2465439, 3679263, 4286175, 6327871, 7348719
Offset: 0

Views

Author

N. J. A. Sloane, Mar 02 2023

Keywords

Examples

			n=4: a(4) = A120963(4) + A341711(2) = 24 + 19 = 43.

Links

Crossrefs

Cf. A120963, A341711.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; nops(invphi(n)) end:
g:= proc(n) option remember; `if`(n=0, 1, add(
      g(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
    end:
a:= n-> g(n)+g(2*iquo(n, 2)+1)/2:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 02 2023
  • Mathematica
    b[n_] := b[n] = Length[invphi[n]];
g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[d*b[d], {d, Divisors[j]}], {j, 1, n}]/n];
a[n_] := g[n] + g[2*Quotient[n, 2] + 1]/2;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 16 2023, after Alois P. Heinz and using Maxim Rytin's invphi program (see A007617) *)

A361151 a(n) = K(n-1) + K(n) + K(n+1), where K(n) = A341711(floor(n/2)).

Original entry on oeis.org

2, 7, 11, 29, 43, 97, 137, 283, 389, 749, 1003, 1839, 2421, 4259, 5515, 9391, 12011, 19887, 25143, 40665, 50931, 80679, 100161, 155847, 192051, 294047, 359839, 543127, 660623, 984239, 1190359, 1752799, 2109119, 3072351, 3679263, 5307023, 6327871, 9044395
Offset: 0

Views

Author

N. J. A. Sloane, Mar 02 2023

Keywords

Examples

			n=4: 5+19+19 = 43 = a(4).

Crossrefs

Cf. A341711.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; nops(invphi(n)) end:
g:= proc(n) option remember; `if`(n=0, 1, add(
      g(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
    end:
a:= n-> add(g(2*floor((i+n)/2)+1)/2, i=-1..1):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 02 2023
  • Mathematica
    nmax1 = 40;
terms = nmax1 + 2; (* number of terms of A120963 *)
nmax2 = Floor[terms/2] - 1;
S[m_] := S[m] = CoefficientList[Product[1/(1 - x^EulerPhi[k]), {k, 1, m*terms}] + O[x]^(terms + 1), x];
S[m = 1]; S[m++]; While[S[m] != S[m - 1], m++];
A120963 = S[m];
A341711[n_ /; 0 <= n <= nmax2] := A120963[[2 n + 2]]/2;
K[n_] := A341711[Floor[n/2]];
a[n_] := If[n == 0, 2, K[n - 1] + K[n] + K[n + 1]];
Table[a[n], {n, 0, nmax1}] (* Jean-François Alcover, Dec 01 2023 *)

A120963 Number of monic polynomials with integer coefficients of degree n with all roots on the unit circle; number of products of cyclotomic polynomials of degree n.

Original entry on oeis.org

1, 2, 6, 10, 24, 38, 78, 118, 224, 330, 584, 838, 1420, 2002, 3258, 4514, 7134, 9754, 15010, 20266, 30532, 40798, 60280, 79762, 115966, 152170, 217962, 283754, 401250, 518746, 724866, 930986, 1287306, 1643626, 2250538, 2857450, 3878298, 4899146, 6594822
Offset: 0

Views

Author

Franklin T. Adams-Watters, Jul 19 2006

Keywords

Comments

Also the number of types of crystallographic rotations and reflection-rotations in n-dimensional Euclidean space. - Andrey Zabolotskiy, Jul 08 2017

Examples

			The six polynomials of degree 2 consist of 3 irreducible cyclotomic polynomials: x^2+1, x^2+x+1 and x^2-x+1 and 3 products of 2 linear cyclotomic polynomials: x^2+2x+1, x^2-1 and x^2-2x+1.
The six plane crystallographic operations are the identity operation, rotations by 2 Pi/k with k = 2,3,4,6, and a reflection.

References

  • Boyd, David W.(3-BC); Montgomery, Hugh L.(1-MI), Cyclotomic partitions. In Number theory (Banff, AB, 1988), 7-25. Walter de Gruyter & Co., Berlin, 1990 ISBN:3-11-011723-1, MR1106647. [Asymptotics]

Links

Crossrefs

Cf. A014197, A051894, A280611 (variant where repeated roots are not allowed).
See also A341710, A341711, A341712.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; nops(invphi(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 04 2019
  • Mathematica
    terms = 40;
S[m_] := S[m] = CoefficientList[Product[1/(1 - x^EulerPhi[k]),
     {k, 1, m*terms}] + O[x]^terms, x];
S[m = 1];
S[m++];
While[S[m] != S[m-1], m++];
S[m] (* Jean-François Alcover, Apr 14 2017, after Christopher J. Smyth, updated May 13 2022 *)

Formula

Euler transform of A014197.
G.f.: Product_{k>=1} 1/(1-x^phi(k)) = Product_{j>=1} (1-x^j)^(-A014197(j)). - Christopher J. Smyth, Jan 08 2017
log(a(n)) ~ sqrt(105*zeta(3)*n)/Pi. - Vaclav Kotesovec, Sep 02 2021

A341710 a(n) = A120963(n)/2.

Original entry on oeis.org

1, 3, 5, 12, 19, 39, 59, 112, 165, 292, 419, 710, 1001, 1629, 2257, 3567, 4877, 7505, 10133, 15266, 20399, 30140, 39881, 57983, 76085, 108981, 141877, 200625, 259373, 362433, 465493, 643653, 821813, 1125269, 1428725, 1939149, 2449573, 3297411, 4145249, 5538254
Offset: 1

Views

Author

N. J. A. Sloane, Feb 19 2021

Keywords

Crossrefs

Cf. A120963, A341711, A341712.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; nops(invphi(n)) end:
a:= proc(n) option remember; `if`(n=0, 1/2, add(
      a(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 19 2021
  • Mathematica
    terms = 40;
S[m_] := S[m] = CoefficientList[Product[1/(1 - x^EulerPhi[k]),
     {k, 1, m*terms}] + O[x]^(terms+1), x]/2 // Rest;
S[m = 1];
S[m++];
While[S[m] != S[m-1], m++];
S[m] (* Jean-François Alcover, May 12 2022 *)

A341712 a(n) = A120963(2*n)/2.

Original entry on oeis.org

3, 12, 39, 112, 292, 710, 1629, 3567, 7505, 15266, 30140, 57983, 108981, 200625, 362433, 643653, 1125269, 1939149, 3297411, 5538254, 9195371, 15104245, 24561098, 39562657, 63160404, 99987453, 157029090, 244754385, 378754786, 582124254, 888874067, 1348842728
Offset: 1

Views

Author

N. J. A. Sloane, Feb 19 2021

Keywords

Comments

A bisection of A341710.

Crossrefs

Cf. A120963, A341710, A341711.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; nops(invphi(n)) end:
g:= proc(n) option remember; `if`(n=0, 1, add(
      g(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
    end:
a:= n-> g(2*n)/2:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 19 2021
  • Mathematica
    terms = 64; (* number of terms of A120963 *)
nmax = Floor[terms/2];
S[m_] := S[m] = CoefficientList[Product[1/(1 - x^EulerPhi[k]),
     {k, 1, m*terms}] + O[x]^(terms+1), x];
S[m = 1];
S[m++];
While[S[m] != S[m-1], m++];
A120963 = S[m];
a[n_ /; 1 <= n <= nmax] := A120963[[2n+1]]/2;
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, May 12 2022 *)
Showing 1-5 of 5 results.

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

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