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A138796
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Least possible k > 0 with T(k) - T(j) = n, j > 0, where T(i) > 0 are the triangular numbers A000217.
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5
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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
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2,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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T=#(#+1)/2&; Min[k/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#, 0<j<k}, {j, k}, Integers]]}]&/@Range[2, 100]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Peter Pein (petsie(AT)dordos.net), Mar 30 2008
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STATUS
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approved
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