A225400 -id:A225400 - cp's OEIS frontend

A225399 Number of nontrivial triangular numbers dividing triangular(n).

0, 0, 0, 1, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 3, 4, 0, 1, 1, 1, 6, 2, 0, 2, 4, 0, 1, 3, 0, 2, 2, 0, 3, 1, 0, 8, 2, 0, 1, 5, 1, 2, 2, 0, 7, 3, 0, 2, 4, 0, 2, 3, 0, 1, 4, 3, 4, 1, 0, 4, 4, 0, 2, 5, 1, 3, 1, 0, 2, 4, 0, 3, 3, 0, 2, 5, 0, 4, 1, 1, 7, 1, 0, 3, 8, 0, 1

Offset: 0

Alex Ratushnyak, May 06 2013

nonn

Number of triangular numbers t such that t divides triangular(n), and 1 < t < triangular(n).

			triangular(3) = 6 is divisible by triangular(2) = 3, so a(3) = 1.
triangular(8) = 36 is divisible by triangular(2) = 3 and triangular(3) = 6, so a(8) = 2.

Cf. A000217, A225400.

Cf. A076982, A076983, A084260, A137281, A141283.

#include 
int main() {
  unsigned long long c, i, j, t, tn;
  for (i = tn = 0; i < (1ULL<<32); i++) {
        for (c=0, tn += i, t = j = 3; t*2 <= tn; t+=j, ++j)
                if (tn % t == 0)  ++c;
        printf("%llu, ", c);
  }
  return 0;
}

A225399 := proc(n)
    option remember ;
    local a,tn,i;
    a := 0 ;
    tn := A000217(n) ;
    for i from 2 to n-1 do
        if modp(tn,A000217(i))=0 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A225399(n),n=0..80) ; # R. J. Mathar, Jan 12 2024

tri = Table[n (n + 1)/2, {n, 100}]; Table[cnt = 0; Do[If[Mod[tri[[n]], tri[[k]]] == 0, cnt++], {k, 2, n - 1}]; cnt, {n, 0, Length[tri]}] (* T. D. Noe, May 07 2013 *)

a(n) = A076982(n) - 2 for n > 1.

A368855 Index of the first occurrence of n in A076982.

Original entry on oeis.org

1, 2, 3, 8, 14, 15, 39, 20, 44, 35, 195, 119, 104, 594, 224, 384, 455, 539, 440, 560, 3080, 2184, 1539, 2015, 2639, 5264, 4199, 15399, 13299, 8855, 23919, 2079, 30744, 43680, 36575, 14399, 5984, 58695, 113399, 47124, 107184, 12375, 78624, 98175, 73359, 111320, 242879, 185724

Offset: 1

Robert G. Wilson v, Jan 09 2024

nonn

			a(1) = 1 since A076982(1) = 1;
a(2) = 2 since A076982(2) = 2;
a(3) = 3 since A076982(3) = 3;
a(4) = 8 since A076982(8) = 4;
a(5) = 14 since A076982(14) = 5; etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..155

Cf. A000217, A002378, A076982, A076983.

Essentially the same as A225400.

k = 1; t[] := 0; f[n] := Length@ Select[ Divisors[n (n +1)/2], IntegerQ@ Sqrt[8# +1] &]; While[k < 250000, a = f@k; If[ t[a] == 0, t[a] = k]; k++]; t /@ Range@ 48

a(n) = (sqrt(8*A076983(n)+1)-1)/2. - Amiram Eldar, Jan 10 2024

cp's OEIS Frontend

A225399 Number of nontrivial triangular numbers dividing triangular(n).

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