A138796 Least possible k > 0 with T(k) - T(j) = n, j > 0, where T(i) > 0 are the triangular numbers A000217.

2, 3, 4, 3, 6, 4, 8, 4, 10, 6, 5, 7, 5, 6, 16, 9, 6, 10, 6, 8, 7, 12, 9, 7, 8, 7, 28, 15, 8, 16, 32, 8, 10, 8, 13, 19, 11, 9, 10, 21, 9, 22, 9, 10, 13, 24, 17, 10, 12, 11, 10, 27, 10, 13, 11, 12, 16, 30, 11, 31, 17, 11, 64, 11, 18, 34, 12, 14, 13, 36, 12, 37, 20, 12, 13, 12, 21, 40, 18

Offset

2,1

Comments

For T(k) see A138797, for j see A138798 and for T(j) see A138799.

The number of ways n can be written as difference of two triangular numbers is sequence A136107

Note that n = t(k)-t(j) implies 2n = (k-j)(k+j+1), where (k-j) and (k+j+1) are of opposite parity. Let d be the odd element of { k-j, k+j+1 }. Then d is an odd divisor of n and k = ( d + 2n/d - 1 ) / 2. Therefore a(n) = ( min{ d + 2n/d } - 1 ) / 2 where d runs through all odd divisors of n, except perhaps (sqrt(8*n+1) +- 1)/2 which correspond to j=0. See PARI program. The restriction that j > 0 seems artificial. If it is removed we get A212652. - Max Alekseyev, Mar 31 2008

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Peter Pein, Mathematica notebook containing a faster algorithm.

Example

a(30)=8, because 30 = T(30) - T(29) = T(11) - T(8) = T(9) - T(5) = T(8) - T(3) and 8 is the least index of the minuends.

Mathematica

T=#(#+1)/2&; Min[k/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#, 0<j<k}, {j, k}, Integers]]}]&/@Range[2, 100]

Prog

(PARI) { a(n) = local(m); m=2*n+1; fordiv(n/2^valuation(n, 2), d, if((2*d+1)^2!=8*n+1&&(2*d-1)^2!=8*n+1, m=min(m, d+(2*n)\d))); (m-1)\2 }
vector(100, n, a(n)) \\ Max Alekseyev, Mar 31 2008

Crossrefs

Cf. A000217, A109814, A118235, A136107, A138797, A138798, A138799, A212652.

Sequence in context: A088043 A332931 A248376 * A186970 A064380 A290732

Adjacent sequences: A138793 A138794 A138795 * A138797 A138798 A138799

Keyword

nonn

Author

Peter Pein (petsie(AT)dordos.net), Mar 30 2008

Status

approved