cp's OEIS Frontend

The name of this constant was suggested by Finch (2003).

Gaussian twin primes on the line x + i in the complex plane are Gaussian primes pair of the form (m - 1 + i, m + 1 + i). The numbers m are numbers such that (m-1)^2 + 1 and (m+1)^2 + 1 are both primes (A096012 plus 1).

Shanks (1960) conjectured that the number of these pairs with m <= x is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section and evaluated it by 0.4876.

The first 100 digits of 4*c were calculated by Ettahri et al. (2019).

4*n*(k + n) is a square. If n is a square, then k + n is also a square.

If n is prime, then n divides k.

If we add the additional condition that p and q are two consecutive primes of the form m^2 + 1, then we obtain the sequence A339008, with A339008(n) = a(n) for n = 1, 2, 3, 4, 6, 7 and 9.

4*n*(k + n) is a square. If n is a square, then k + n is also a square.

If n is prime, then n divides k.

a(n) = A339007(n) for n = 1, 2, 3, 4, 6, 7 and 9.

The sequence is probably infinite. [This follows from Schinzel's hypothesis H, for example. - Charles R Greathouse IV, Aug 31 2021]

A majority of numbers in the sequence are of the form 2*q^2 with q = 5, 25, 30, 35, 70, 85, 100, 110, 225, 230, 260, 285, 290, 320, 390, 410, ... So, it seems that {a(n)} = {334, 516, 670, 844, 1164} union {2*A109306(n)^2} where A109306 are the numbers k such that k^2 + (k-1)^2 and k^2 + (k+1)^2 are both primes.

Or number of integers of the form m^2+1 between two consecutive twin k^2+1 primes.

a(n)==7 mod 10 for n>1.

The primes of the sequence are 7, 17, 37, … (A030432), subsequence of A017353.

Conjecture 1: the sequence is infinite.

Conjecture 2: for n>1, the sequence sorted in ascending order with distinct values generates the set B = {b(k)} = {7, 17, 27, 37, …} = {7 + 10k}, k = 0,1,2,… and {a(n)}/qZ = B/qZ with q = 2^m, m = 1, 2,… is a multiplicative group.

This has been verified for n up to 10^8.

A remarkable simple way to perceive properties of invariability in this sequence (or other particular sequences) is the use of finite groups of integers modulo q. This study can provide interesting interpretations for some regularities which describe properties in other mathematical spaces.

However, this concept is based on a conjecture if it is impossible to prove the infinity character of a sequence. This requires, for the calculations, a large number of elements in the sequence.

The groups of integers modulo 2^m are:

q = 2 => B/2Z = {1}, the trivial group.

q = 4 => B/4Z = {1,3}, the cyclic group of order 2 with two elements.

q = 8 => B/8Z = {1,3,5,7}, the group of order 4 with generating set {3,7} => the Klein four-group of order 2. The square of each element of B/8Z is 1. The group is not cyclic.

q = 16 => B/16Z = {1,3,5,7,9,11,13,15}, the group of order 8 with generating set {3,15}. The powers of 3 {1,3,9,11} are a subgroup of order 4, as are the powers of 5, {1,5,9,13}. The group B/16Z is not cyclic.

For higher powers q = 2^k, k>2, B/(2^k)Z = {1,3,5,…,2^k-1}, with generating set {3, 2^k-1}. The group B/(2^k)Z is not cyclic.

The order of the group is given by Euler’s totient function (A000010): this is the product of the orders of the cyclic groups in the direct product (see the links).

Or decimal expansion of (1/5 + 1/17) + Sum_{i>=0} (1/p(i) + 1/q(i)) where p(i) and q(i) are primes of the form p(i) = m^2 + 1 = (10*i+4)^2 + 1 and q(i) = (m + 2)^2 + 1 = (10*i + 6)^2 + 1 (for m > 1, m == 4 (mod 10)). See A096012.

The sum is convergent; it must be less than 0.81459657... (see A172168).

Conjecture: the series of all twin m^2 + 1 prime reciprocals converges to 0.357745147...

It is probable that a(9) = 1.

A good approximation to the constant is (2*log(7/3)/log(17))^2 = 0.35774506... which agrees with the constant through the first 6 significant digits.