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%I #98 Dec 23 2020 08:42:19
%S 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,
%T 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,
%U 11,1,1,6,2,1,3,2,2,3,1,2,6,1
%N Numerators in a set of expansions of the single-term Machin-like formula for Pi.
%C Denominators are A338879.
%C 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:
%C k= 1 2 3 4 5 6
%C n=1: 1;
%C n=2: 1/3, 1/2;
%C n=3: 1/5, 3/11, 1/3;
%C n=4: 1/7, 2/11, 2/9, 1/4;
%C n=5: 1/9, 5/37, 5/31, 5/27, 1/5;
%C n=6: 1/11, 3/28, 1/8, 1/7, 3/19, 1/6;
%C 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.
%H Sanjar Abrarov, <a href="/A338878/b338878.txt">Table of n, a(n) for n = 1..120</a>
%H Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, and Brendan M. Quine, <a href="https://arxiv.org/abs/2004.11711">Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi</a>, arXiv:2004.11711 [math.GM], 2020.
%F 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.
%F Pi/4 = Sum_{k=1..n} arctan(T(n,k) / A338879(n,k)).
%e Triangle T(n,k) begins:
%e k= 1 2 3 4 5 6
%e n=1: 1;
%e n=2: 1, 1;
%e n=3: 1, 3, 1;
%e n=4: 1, 2, 2, 1;
%e n=5: 1, 5, 5, 5, 1;
%e n=6: 1, 3, 1, 1, 3, 1;
%e 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.
%e 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.
%t (*Define variable*)
%t PiOver4[m_] := Sum[ArcTan[m/((k - 1)*k + m^2)], {k, 1, m}];
%t (*Expansions*)
%t m := 1;
%t While[m <= 10,
%t If[m == 1, Print["\[Pi]/4 = ArcTan[1/1]"],
%t Print["\[Pi]/4 = ", PiOver4[m]]]; m = m + 1];
%t (*Verification*)
%t m := 1;
%t While[m <= 10, Print[PiOver4[m] == Pi/4]; m = m + 1];
%t (*Numerators*)
%t For[n = 1, n <= 15, n++, {k := 1; sq := {};
%t While[n >= k, AppendTo[sq, n/GCD[n, k*(k - 1)]]; k++]};
%t Print[sq]];
%o (PARI) T(n, k) = if (n>=k, n/gcd(n,k*(k - 1)))
%o matrix(10, 10, n, k, T(n, k)) \\ _Michel Marcus_, Nov 14 2020
%Y Cf. A338879 (denominators), A003881 (Pi/4).
%K nonn,frac,tabl
%O 1,5
%A _Sanjar Abrarov_, Nov 13 2020
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