A212652 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.

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

Offset: 1

Author

L. Edson Jeffery, Feb 14 2013

Keywords

• nonn

Comments

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

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

From Daniel Forgues, Jan 06 2016: (Start)

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

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.)

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)

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

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

Conjecture: a(n) is the smallest of the largest parts of the partitions of n into consecutive parts. - Omar E. Pol, Jan 07 2025

Examples

			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.
		

Links

• David W. Wilson, Table of n, a(n) for n = 1..10000
• Max Alekseyev, is this sequence interesting?, Sequence Fans Mailing List, Mar 31 2008.

Crossrefs

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

Programs

Maple

f:= n ->  min(map(t -> n/t + (t-1)/2,
numtheory:-divisors(n/2^padic:-ordp(n,2)))):
map(f, [$1..100]); # Robert Israel, Jan 14 2016

Mathematica

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

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

Formula

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

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

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

Extensions

Reference to Max Alekseyev's 2008 proposal of this sequence added by N. J. A. Sloane, Nov 01 2014