A242976 Number of ways to write n = i + j + k with 0 < i <= j <= k such that prime(i) mod i, prime(j) mod j and prime(k) mod k are all triangular numbers, where prime(m) mod m denotes the least nonnegative residue of prime(m) modulo m.

0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 8, 8, 9, 8, 9, 7, 9, 9, 9, 8, 9, 8, 9, 9, 10, 9, 9, 8, 8, 9, 9, 9, 8, 9, 9, 11, 13, 12, 12, 13, 13, 13, 12, 14, 12, 11, 13, 10, 13, 11, 15, 12, 12, 13, 11, 12, 11, 9, 8, 7, 9, 11, 12, 15, 12, 16, 17, 15, 19, 15, 19, 12, 17, 15, 15, 17, 15

Offset

1,6

Comments

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) Any integer n > 3 can be written as a + b + c + d with a, b, c, d in the set {k>0: prime(k) mod k is a square}.

Clearly, part (i) implies that there are infinitely many positive integer k with prime(k) mod k a triangular number, and part (ii) implies that there are infinitely many positive integer k with prime(k) mod k a square.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..6000

Example

a(5) = 1 since 5 = 1 + 2 + 2, prime(1) == 0*1/2 (mod 1) and prime(2) = 3 == 1*2/2 (mod 2). Note that prime(3) = 5 == 2 (mod 3) with 2 not a triangular number.

Mathematica

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
t[k_]:=TQ[Mod[Prime[k], k]]
a[n_]:=Sum[Boole[t[i]&&t[j]&&t[n-i-j]], {i, 1, n/3}, {j, i, (n-i)/2}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A000217, A000290, A000720, A242950.

Sequence in context: A255572 A065855 A236863 * A218445 A034137 A156351

Adjacent sequences: A242973 A242974 A242975 * A242977 A242978 A242979

Keyword

nonn

Author

Zhi-Wei Sun, May 28 2014

Status

approved