

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 rth triangular number.


6



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
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OFFSET

1,2


COMMENTS

Conjecture: n appears in row a(n) of A209260.
n = Sum_{i=k+1..M} i = T(M)  T(k) = (Mk)*(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 = M2. 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 + (p1)/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



FORMULA

a(n) = Min_{odd dn} (n/d + (d1)/2).


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(119) = T(11)  T(2) = 66  3 = 63.


MAPLE

f:= n > min(map(t > n/t + (t1)/2,
numtheory:divisors(n/2^padic:ordp(n, 2)))):


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)); (m1)\2; } \\ Max Alekseyev, Mar 31 2008


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



