A283190 - OEIS (original) (raw)

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

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

EXAMPLE

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

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:

MATHEMATICA

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

PROG

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