A289777 Frequency of the largest spectral component of the prime characteristic function of the first n numbers, for n>3, excluding the smallest and largest frequencies. In case of a tie, use the smallest frequency.

2, 2, 3, 3, 4, 3, 5, 5, 4, 6, 6, 4, 7, 7, 7, 8, 8, 5, 9, 9, 9, 6, 6, 6, 6, 11, 11, 7, 12, 12, 7, 13, 13, 14, 14, 14, 15, 15, 15, 8, 16, 16, 17, 17, 17, 11, 18, 18, 10, 10, 19, 12, 20, 20, 11, 11, 21, 21, 22, 22, 12, 23, 23, 12, 24, 24, 13, 25, 25, 13, 26, 26, 14, 27, 27, 14, 28, 28, 29, 29, 29, 15, 30, 30, 16, 16

Offset: 4

Author

Andres Cicuttin, Jul 12 2017

Keywords

• nonn

Comments

The Discrete Fourier transform is applied to the list of the prime characteristic function (A010051) of the first n numbers; then the position of the largest absolute value of the components of the transformed list, disregarding the first and last components, is selected. If there are several identical maxima then the lowest position of them is taken.

The scatter plot of these maximum spectral components exhibits a curious pattern in which these components are essentially aligned along two convergent directions (see link).

It seems that the Fourier spectrum of the prime characteristic function is remarkably symmetric when obtained from a list with an even numer of elements (see link) and it could be related to the symmetry found in the distribution of consecutive and alternate primes gap ratios (see comments and plots in A274263 and A276309).

Conjecture: lim_{n->inf} abs(4a(n)/n - 1) = 1/3.

Examples

			For the first 43 terms of the characteristic function of primes (A010051), the absolute values of its discrete Fourier transform have a maximum at position 8 after excluding the smallest frequency (first position) and the largest frequency (last position), then a(43) = 8.
		

Links

• Andres Cicuttin, Scatter plot of first 2^13 terms
• Andres Cicuttin, Fourier spectrum of first 2^16 values of the characteristic function of primes (A010051), excluding the first and last frequency components.
• A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004; Amer. Math. Monthly vol 113, no 1 (2006) p 1.

Crossrefs

Cf. A010051, A273784, A274263, A276309.

Programs

Mathematica

PrimeChar[n_] := If[PrimeQ[n] == True, 1, 0];
Table[Position[b = Abs@Fourier@Table[PrimeChar[j], {j, 1, n}],
   Max[b[[2 ;; Floor[n/2]]]]][[1, 1]], {n, 4, 160}]