A320044 Number of positive integers k < prime(n)/2 with {k^3/prime(n)} > 1/2, where {x} = x - floor(x) is the fractional part of a real number x.

0, 0, 1, 1, 2, 4, 3, 2, 4, 7, 9, 10, 7, 9, 12, 15, 12, 14, 15, 18, 19, 25, 18, 19, 20, 23, 28, 22, 30, 27, 34, 30, 33, 37, 39, 35, 40, 38, 36, 51, 38, 42, 50, 50, 49, 53, 44, 57, 62, 59, 55, 54, 49, 62, 65, 62, 69, 59, 65, 67, 77, 69, 71, 80, 80, 69, 76, 78, 88, 87, 87, 94, 87, 87, 99, 96, 87, 97, 97, 94

Offset: 1

Author

Zhi-Wei Sun, Oct 04 2018

Keywords

• nonn

Comments

Conjecture 1: For any prime p == 5 (mod 6), the difference card{0 < k < p/2: {k^3/p} > 1/2} - (p+1)/6 is nonnegative and even.

Conjecture 2: For any prime p not congruent to 1 modulo 5, the number of positive integers k < p/2 with {k^5/p} > 1/2 is even.

Conjecture 3: For any prime p == 5 (mod 12), the difference card{0 < k < p/2: {k^6/p} > 1/2} - (p-5)/12 is positive and odd.

Examples

			a(3) = 1 since prime(3) = 5 and {0 < k < 5/2: {k^3/5} > 1/2} = {2}.
a(4) = 1 since prime(4) = 7 and {0 < k < 7/2: {k^3/7} > 1/2} = {3}.
a(5) = 2 since prime(5) = 11 and {0 < k < 11/2: {k^3/11} > 1/2} = {2,4}.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Quadratic residues and related permutations and identities, arXiv:1809.07766 [math.NT], 2018.

Crossrefs

Cf. A000040, A000578, A000584, A001014.

Programs

Mathematica

s[p_]:=s[p]=Sum[Boole[Mod[k^3,p]>p/2],{k,1,(p-1)/2}];Table[s[Prime[n]],{n,1,80}]