Moritz Firsching - cp's OEIS frontend

A263183 Decimal expansion of 1/log_2(r), where r is Otter's rooted tree constant.

6, 3, 9, 5, 7, 7, 6, 8, 9, 9, 9, 4, 7, 2, 0, 1, 3, 3, 1, 1, 2, 8, 9, 9, 8, 7, 0, 5, 6, 5, 7, 3, 1, 3, 8, 4, 1, 1, 5, 2, 7, 6, 4, 8, 1, 9, 1, 4, 4, 1, 9, 6, 2, 2, 5, 8, 2, 7, 4, 2, 3, 5, 5, 8, 3, 6, 1, 3, 2, 3, 5, 3, 1, 8, 5, 8, 8, 1, 6, 7, 7, 3, 6, 8, 6, 9, 5, 7, 0, 5, 0, 8, 4, 0, 1, 7, 9, 5, 9, 1

Offset: 0

Moritz Firsching, Oct 11 2015

The article "Beweisbar oder nicht? Die Grenzzahl 0,639578175..." (linked below) has a wrong value (already in the title).

			0.63957768999472013311289987056573138411527648191441962258274235583613235318588...

Andreas Weiermann, An Application of Graphical Enumeration to PA, The Journal of Symbolic Logic, Vol. 68, No. 1 (2003), 5-16 (see page 13).
Andreas Weiermann and Dirk Huylebrouck, Beweisbar oder nicht? Die Grenzzahl 0,639578175..., Mitteilungen der DMV, Volume 23, Issue 3 (2015), 156-159.

Cf. A051491.

Equals 1/log_2(A051491).

A262322 The number of 4-connected triangulations of the triangle with n inner vertices.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 13, 47, 217, 1041, 5288, 27844, 150608, 831229

Offset: 0

Moritz Firsching, Sep 18 2015

Also the number of 4-connected simplicial polyhedra with n nodes with one marked face.

Values obtained by generating 4-connected simplicial polyhedra with plantri, marking each face in the polyhedron, and then sorting out isomorphic ones.

Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]

Cf. A002713, A007021.

A260940 a(n) is the smallest index j>n such that g(j)=0 for the sequence g defined (for indices > n) by g(n+1)=n and g(i) = g(i-1) - gcd(i,g(i-1)).

Original entry on oeis.org

3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 19, 21, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 43, 43, 53, 43, 41, 59, 61, 61, 61, 67, 67, 71, 73, 73, 71, 79, 79, 83, 79, 79, 89, 79, 79, 79, 97, 97, 101, 103, 103, 107, 109, 109, 113, 109, 109, 113, 109

Offset: 1

Moritz Firsching, Aug 04 2015

nonn

a(n) is prime for all n<=10^10 except a(13)=21.
a(n) <= 2n + 1.
a(n) = 2n + 1 if and only if 2n + 1 is prime.
a(n) = 2n - 1 if and only if 2n - 1 is a prime and 2n - 1 = 1 mod 6.
a(n) = 2n - 3 if and only if 2n - 3 is a prime and 2n - 3 = 1 mod 30.

Moritz Firsching, Table of n, a(n) for n = 1..9999

Cf. A002476, A132230, A261301.

A186253(n) is a^n(2) where a^n denotes the n-th composition.

a(last_a) = {
  local(A=last_a,B=last_a,C=2*last_a+1);
  while(A>0,
    D=divisors(C);
    k1=10*D[2];
    for(j=2,matsize(D)[2],d=D[j];k=((A+1-B+d)/2)%d;
      if(k==0,k=d); if(k<=k1,k1=k;d1=d));
    if(k1-1+d1==A,B=B+1);
    A = max(A-(k1-1)-d1,0);
    B = B + k1;
    C = C - (d1 - 1);
  );
  return(B);
}
a(n)={
my(A=n, B=n, C=2*n+1);
while(A>0,
my(k1=oo,d1);
fordiv(C,d,
if(d==1,next);
my(k=((A+1-B+d)/2)%d);
if(k==0, k=d);
if(k<=k1, k1=k; d1=d)
);
A -= k1 - 1 + d1;
B += k1 + (A==0);
C -= d1 - 1;
);
B;
} \\ Charles R Greathouse IV, Nov 04 2015

def a(n):
    g=n
    n+=1
    while(g!=0):
        g=g-gcd(n,g)
        n+=1
    return n

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User: Moritz Firsching

A263183 Decimal expansion of 1/log_2(r), where r is Otter's rooted tree constant.

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A262322 The number of 4-connected triangulations of the triangle with n inner vertices.

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A260940 a(n) is the smallest index j>n such that g(j)=0 for the sequence g defined (for indices > n) by g(n+1)=n and g(i) = g(i-1) - gcd(i,g(i-1)).

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