A338879 -id:A338879 - cp's OEIS frontend

1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 5, 5, 5, 1, 1, 3, 1, 1, 3, 1, 1, 7, 7, 7, 7, 7, 1, 1, 4, 4, 2, 2, 4, 4, 1, 1, 9, 3, 3, 9, 3, 3, 9, 1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 1, 6, 2, 1, 3, 2, 2, 3, 1, 2, 6, 1

Offset: 1

Sanjar Abrarov, Nov 13 2020

Denominators are A338879.
Abrarov et al. (see section LINKS) give an identity arctan(n*x) = Sum_{m=1..n} arctan(x/(1 + (m-1)*m*x^2)). At x=1/n this identity provides set of expansions of the single-term Machin-like formula for Pi in form Pi/4 = arctan(1) = Sum_{m=1..n} arctan(n/((m-1)*m + n^2)). For m = n - k + 1 at k=1..n the fractions n/((m-1)*m + n^2) constitute the triangle with rows in ascending order:
    k=  1     2     3     4     5     6
  n=1:  1;
  n=2:  1/3,  1/2;
  n=3:  1/5,  3/11, 1/3;
  n=4:  1/7,  2/11, 2/9,  1/4;
  n=5:  1/9,  5/37, 5/31, 5/27, 1/5;
  n=6:  1/11, 3/28, 1/8,  1/7,  3/19, 1/6;
Section EXAMPLE shows the corresponding triangle T(n,k) with numerators. This triangle T(n,k) possesses a sifting property for primes. In particular, in a row of kind {1,p,p,p,...,p,p,1} the integer p must be a prime and n = p except the case n = 4 when prime p = 2.

			Triangle T(n,k) begins:
     k= 1  2  3  4  5  6
  n=1:  1;
  n=2:  1, 1;
  n=3:  1, 3, 1;
  n=4:  1, 2, 2, 1;
  n=5:  1, 5, 5, 5, 1;
  n=6:  1, 3, 1, 1, 3, 1;
For example, for row n = 3 the corresponding expansion formula is Pi/4 = arctan(1/5) + arctan(3/11) + arctan(1/3) and the numerators are 1,3,1.
At n = 3, n = 4 and n = 5 the rows are {1,3,1}, {1,2,2,1} and {1,5,5,5,1} and the primes are 3, 2 and 5, respectively.

Sanjar Abrarov, Table of n, a(n) for n = 1..120
Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, and Brendan M. Quine, Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi, arXiv:2004.11711 [math.GM], 2020.

Cf. A338879 (denominators), A003881 (Pi/4).

(*Define variable*)
PiOver4[m_] := Sum[ArcTan[m/((k - 1)*k + m^2)], {k, 1, m}];
(*Expansions*)
m := 1;
While[m <= 10,
  If[m == 1, Print["\[Pi]/4 = ArcTan[1/1]"],
    Print["\[Pi]/4 = ", PiOver4[m]]]; m = m + 1];
(*Verification*)
m := 1;
While[m <= 10, Print[PiOver4[m] == Pi/4]; m = m + 1];
(*Numerators*)
For[n = 1, n <= 15, n++, {k := 1; sq := {};
  While[n >= k, AppendTo[sq, n/GCD[n, k*(k - 1)]]; k++]};
    Print[sq]];

T(n, k) = if (n>=k, n/gcd(n,k*(k - 1)))
matrix(10, 10, n, k, T(n, k)) \\ Michel Marcus, Nov 14 2020

T(n,k) = numerator(n/((n-k)*(n-k+1) + n^2)) = n/gcd(n,k*(k-1)), for n>=1 and 1 <= k <= n.

Pi/4 = Sum_{k=1..n} arctan(T(n,k) / A338879(n,k)).

cp's OEIS Frontend

A338878 Numerators in a set of expansions of the single-term Machin-like formula for Pi.

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