

A138796


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


5



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
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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 = (kj)(k+j+1), where (kj) and (k+j+1) are of opposite parity. Let d be the odd element of { kj, 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*d1)^2!=8*n+1, m=min(m, d+(2*n)\d))); (m1)\2 } vector(100, n, a(n)) \\ Max Alekseyev, Mar 31 2008


CROSSREFS

Cf. A000217, A109814, A118235, A136107, A138797, A138798, A138799, A212652.
Sequence in context: A167234 A088043 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



