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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.
2
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
1,5
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.
LINKS
Zhi-Wei Sun, Quadratic residues and related permutations and identities, arXiv:1809.07766 [math.NT], 2018.
EXAMPLE
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}.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 04 2018
STATUS
approved