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A333529 Number of triples [n,k,m] with n <= k <= m satisfying T_n + T_k = T_m, where T_i = i*(i+1)/2 are the triangular numbers. 4
0, 1, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 2, 5, 3, 1, 4, 4, 3, 5, 5, 2, 3, 4, 4, 5, 5, 3, 6, 6, 1, 3, 5, 6, 7, 5, 2, 5, 6, 3, 6, 6, 3, 8, 9, 2, 3, 4, 6, 8, 6, 3, 6, 11, 5, 6, 5, 2, 7, 7, 2, 9, 5, 3, 11, 6, 3, 6, 11, 6, 5, 5, 2, 9, 9, 6, 11, 6, 3, 7, 7, 2, 7, 12, 6, 5, 7, 3, 10, 16, 6, 6, 5, 6, 6, 3, 4, 12, 12, 5, 6, 6, 3, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

a(n)=1 if n is in A068194. - Robert Israel, Apr 03 2020

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

There is a list of all triples (including those with 0 < k < n) with n <= 16 in A309507.

MAPLE

with(numtheory):

A:=[]; M:=150; ct:=Array(0..M, 0):

for n from 1 to M do

TT:=n*(n+1);

dlis:=divisors(TT);

  for d in dlis do

if (d mod 2) = 1 then e := TT/d;

mi:=min(d, e); ma:=max(d, e);

k:=(ma-mi-1)/2; m:=(ma+mi-1)/2;

# skip if k<n

    if k>=n then

    ct[n]:=ct[n]+1;

    lprint(n, k, m);

    fi;

fi;

od:

od:

[seq(ct[n], n=1..M)];

# alternative:

f:= proc(n) local t, t0, r, dmax, divs;

    t:= n*(n+1);

    r:= padic:-ordp(t, 2);

    t0:= t/2^r;

    dmax:= floor((sqrt(8*t+1)-1)/2-n);

    divs:= numtheory:-divisors(t0);

    nops(select(`<=`, divs, dmax)) + nops(select(`<=`, divs, dmax/2^r))

end proc:

map(f, [$1..200]); # Robert Israel, Apr 03 2020

MATHEMATICA

T[n_] := n(n+1)/2;

r[n_] := Reduce[n <= k <= m && T[n] + T[k] == T[m], {k, m}, Integers];

a[n_] := Module[{rn = r[n], r0}, r0 = rn[[0]]; Which[r0 === Or, Length[rn], r0 === And, 1, rn === False, 0, True, Print["error ", n, " ", rn]]];

Array[a, 100] (* Jean-Fran├žois Alcover, Jun 08 2020 *)

CROSSREFS

A309507 counts all triples with k>0.

Cf. A000217, A068194.

Sequence in context: A219644 A193676 A029291 * A022872 A091423 A221914

Adjacent sequences:  A333526 A333527 A333528 * A333530 A333531 A333532

KEYWORD

nonn,look

AUTHOR

N. J. A. Sloane, Mar 31 2020

STATUS

approved

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Last modified August 9 01:35 EDT 2020. Contains 336310 sequences. (Running on oeis4.)