A212652 a(n) = 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

1,2

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

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.

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.

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A205563 A147594 A305425 * A303691 A205678 A128590

Adjacent sequences: A212649 A212650 A212651 * A212653 A212654 A212655

Keyword

nonn

Author

L. Edson Jeffery, Feb 14 2013

Extensions

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

Status

approved