A120963 -id:A120963 - cp's OEIS frontend

A347524 E.g.f.: exp(Sum_{k>=1} A014197(k)*x^k).

1, 2, 10, 44, 364, 2552, 28504, 267920, 3762448, 44426528, 733803424, 10281376448, 197599119040, 3231560909696, 69960323019136, 1295278340380928, 31334430312038656, 650842176105505280, 17337350481203210752, 397746862137852603392, 11579274068100022660096

Offset: 0

Vaclav Kotesovec, Sep 05 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A000010, A014197, A120963, A280611.

E.g.f.: exp(Sum_{k>=1} x^A000010(k)).

log(a(n)/n!) ~ 3*sqrt(70*zeta(3)*n)/Pi^2.

A242848 Number of unimodular roots of the equation z^n + z^k - 1 for all 1 <= k <= n-1.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 0, 4, 4, 0, 4, 4, 6, 0, 6, 0, 6, 8, 6, 8, 8, 0, 16, 8, 0, 8, 10, 12, 10, 0, 12, 12, 32, 0, 12, 12, 12, 16, 14, 12, 14, 16, 18, 16, 16, 0, 28, 32, 18, 16, 18, 0, 54, 16, 18, 20, 20, 24, 20, 20, 18, 0, 62, 24, 22, 24, 24, 64, 24, 0, 24, 24

Offset: 2

Tom Edgar, May 23 2014

nonn

A complex root is called unimodular if it lies on the unit circle.
While there is technically 1 unimodular root for z-1, the offset of this sequence is 2 so that the polynomial in question has three terms.
Let g=gcd(n,k). If 6 divides n/g + k/g, then z^n + z^k - 1 has exactly 2*g unimodular roots each of the form exp(i(Pi/3g + 2*Pi*m/g)), or its conjugate, where 0 <= m <= g-1 (see Theorem 2 from College Math Journal reference).

			The polynomial z^20 + z^4 - 1 has 8 roots (of the 20 possible) lying on the unit circle; moreover, z^20 + z^k - 1 has no roots lying on the unit circle when 1 <= k <= 19 and k != 4. Thus a(20) = 8.

Michael A. Brilleslyper and Lisbeth E. Schaubroeck, Locating Unimodular Roots, College Mathematics Journal, Volume 45, Number 3, May 2014, pp. 162-168(7).

Cf. A120963.

[sum(2*gcd(n,k) for k in [1..n-1] if Integer(n/gcd(n,k)+k/gcd(n,k))%6==0) for n in [2..100]]

a(n) = Sum_{k=1..n-1; 6 divides (n/gcd(n,k) + k/gcd(n,k))} 2*gcd(n,k).

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

N. J. A. Sloane, Mar 02 2023

nonn

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

Cf. A341711.

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

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 *)

cp's OEIS Frontend

A347524 E.g.f.: exp(Sum_{k>=1} A014197(k)*x^k).

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A242848 Number of unimodular roots of the equation z^n + z^k - 1 for all 1 <= k <= n-1.

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

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