A283190 a(n) is the number of different values n mod k for 1 <= k <= floor(n/2).

0, 1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 2, 3, 3, 3, 4, 5, 4, 5, 4, 4, 5, 6, 5, 6, 7, 7, 7, 8, 6, 7, 7, 7, 8, 9, 8, 9, 9, 9, 10, 11, 9, 10, 9, 9, 10, 11, 10, 11, 12, 12, 12, 13, 12, 13, 13, 13, 14, 15, 13, 14, 14, 14, 15, 16, 15, 16, 15, 15, 16, 17, 16, 17, 17, 17, 17, 18, 17

Offset: 1

Author

Thomas Kerscher, Mar 02 2017

Keywords

• nonn
• changed

Comments

a(n) is the number of distinct terms in the first half of the n-th row of the A048158 triangle. - Michel Marcus, Mar 04 2017

a(n)/n appears to converge to a constant, approximately 0.2296. Can this be proved, and does the constant have a closed form? - Robert Israel, Mar 13 2017

The constant that a(n)/n approaches is Sum {p prime} 1/(p^2+p)* Product {q prime < p} (q-1)/q. - Michael R Peake, Mar 16 2017

Examples

			a(7) = 2 because 7=0 (mod 1), 7=1 (mod 2), 7=1 (mod 3), two different results.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Omkar Baraskar and Ingrid Vukusic, Bounds for sets of remainders, arXiv:2508.20853 [math.NT], 2025.
• Michael R Peake, Explanation of limiting value of a(n)/n

Crossrefs

Cf. A048158.

Programs

Maple

N:= 100: # to get a(1)..a(N)
V:= Vector(N,1):
V[1]:= 0:
for m from 2 to N-1 do
  k:= m/min(numtheory:-factorset(m));
  ns:= [seq(n,n=m+1..min(N,m+k-1))];
  V[ns]:= map(`+`,V[ns],1);
od:
convert(V,list); # Robert Israel, Mar 13 2017

Mathematica

Table[Length@ Union@ Map[Mod[n, #] &, Range@ Floor[n/2]], {n, 78}] (* Michael De Vlieger, Mar 03 2017 *)

PARI

a(n) = #vecsort(vector(n\2, k, n % k),,8); \\ Michel Marcus, Mar 02 2017