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a(n) is the least positive integer M such that n = T(M) - T(k), for k an integer, 0 <= k <= M, where T(r) = A000217(r) is the r-th triangular number.
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%I #84 Jan 11 2025 04:02:22

%S 1,2,2,4,3,3,4,8,4,4,6,5,7,5,5,16,9,6,10,6,6,7,12,9,7,8,7,7,15,8,16,

%T 32,8,10,8,8,19,11,9,10,21,9,22,9,9,13,24,17,10,12,11,10,27,10,10,11,

%U 12,16,30,11,31,17,11,64,11,11,34

%N a(n) is the least positive integer M such that n = T(M) - T(k), for k an integer, 0 <= k <= M, where T(r) = A000217(r) is the r-th triangular number.

%C n = A000217(a(n)) - A000217(a(n) - A109814(n)).

%C Conjecture: n appears in row a(n) of A209260.

%C From _Daniel Forgues_, Jan 06 2016: (Start)

%C n = Sum_{i=k+1..M} i = T(M) - T(k) = (M-k)*(M+k+1)/2.

%C n = 2^m, m >= 0, iff M = n = 2^m and k = n - 1 = 2^m - 1. (Points on line with slope 1.) (Powers of 2 can't be the sum of consecutive numbers.)

%C n is odd prime iff k = M-2. Thus M = (n+1)/2 when n is odd prime. (Points on line with slope 1/2.) (Odd primes can't be the sum of more than 2 consecutive numbers.) (End)

%C If n = 2^m*p where p is an odd prime, then a(n) = 2^m + (p-1)/2. - _Robert Israel_, Jan 14 2016

%C This also expresses the following geometry: along a circle having (n) points on its circumference, a(n) expresses the minimum number of hops from a start point, in a given direction (CW or CCW), when each hop is increased by one, before returning to a visited point. For example, on a clock (n=12), starting at 12 (same as zero), the hops would lead to the points 1, 3, 6, 10 and then 3, which was already visited: 5 hops altogether, so a(12) = 5. - _Joseph Rozhenko_, Dec 25 2023

%C Conjecture: a(n) is the smallest of the largest parts of the partitions of n into consecutive parts. - _Omar E. Pol_, Jan 07 2025

%H David W. Wilson, <a href="/A212652/b212652.txt">Table of n, a(n) for n = 1..10000</a>

%H Max Alekseyev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/pipermail/seqfan/2008-March/050941.html">is this sequence interesting?</a>, Sequence Fans Mailing List, Mar 31 2008.

%F a(n) = Min_{odd d|n} (n/d + (d-1)/2).

%F a(n) = A218621(n) + (n/A218621(n) - 1)/2.

%F a(n) = A109814(n) + A118235(n) - 1.

%e For n = 63, we have D(63) = {1,3,7,9,21,63}, B_63 = {11,12,13,22,32,63} and a(63) = min(11,12,13,22,32,63) = 11. Since A109814(63) = 9, T(11) - T(11-9) = T(11) - T(2) = 66 - 3 = 63.

%p f:= n -> min(map(t -> n/t + (t-1)/2,

%p numtheory:-divisors(n/2^padic:-ordp(n,2)))):

%p map(f, [$1..100]); # _Robert Israel_, Jan 14 2016

%t Table[Min[n/# + (# - 1)/2 &@ Select[Divisors@ n, OddQ]], {n, 67}] (* _Michael De Vlieger_, Dec 11 2015 *)

%o (PARI) { A212652(n) = my(m); m=2*n+1; fordiv(n/2^valuation(n,2), d, m=min(m,d+(2*n)\d)); (m-1)\2; } \\ _Max Alekseyev_, Mar 31 2008

%Y Cf. A000217, A109814, A118235, A138796, A141419, A209260, A218621.

%K nonn

%O 1,2

%A _L. Edson Jeffery_, Feb 14 2013

%E Reference to _Max Alekseyev_'s 2008 proposal of this sequence added by _N. J. A. Sloane_, Nov 01 2014