

A111774


Numbers that can be written as a sum of at least three consecutive positive integers.


12



6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102
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OFFSET

1,1


COMMENTS

In this sequence there are no (odd) primes and there are no powers of 2.
So we have only three kinds of natural numbers: the odd primes, the powers of 2 and the numbers that can be represented as a sum of at least three consecutive integers.
Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of k consecutive integers (other than the trivial n=n, for k=1).
Numbers of the form (x*(x+1)y*(y+1))/2 for nonnegative integers x,y with xy >= 3.  Bob Selcoe, Feb 21 2014


REFERENCES

Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC, Problem C, Jun 2005, p. 181182
Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, Solution to problem 3G p. 179.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nieuw Archief voor Wiskunde 5/6 nr. 2 Problems/UWC, Problem C: solution of this Problem
J. Spies, Sage program for computing A111774


EXAMPLE

a(1)=6 because 6 is the first number that can be written as a sum of three consecutive positive integers: 6 = 1+2+3.
Let the top row of an array be A000217(n). Let the diagonals (reading down and left) be A000217(n)A000217(1), A000217(n)A000217(2), A000217(n)A000217(3)..., A000217(n)A000217(n3). This is the square array of A049777, starting with the third column. The array begins as follows:
6 10 15 21 28 36 45 55 66
9 14 20 27 35 44 54 65
12 18 25 33 42 52 63
15 22 30 39 49 60
18 26 35 45 56
21 30 40 51
24 34 45
27 38
30
This is (x*(x+1)y*(y+1))/2 for nonnegative integers x,y with xy>=3, because it is equivalent to 1+2+3/+4/+5/...+x/0/1/2/3/4/5/...(x+3)/ for all possible strings of consecutive integers, which represents every possible way to sum three or more consecutive positive integers. So for example, 4+5+6+7 = 1+2+3+4+5+6+7123 = 22, which is x*(x+1)y*(y+1))/2 when x=7, y=3. Notice that values can appear more than once in the array because some numbers can be represented as sums of more than one string of three or more consecutive positive integers. For example, 30 = x*(x+1)y*(y+1))/2 when (a) x=11, y=8: 9+10+11; (b) x=9, y=5: 6+7+8+9; and (c) x=8, y=3: 4+5+6+7+8. By definition, xy is the number of integers in the string.  Bob Selcoe, Feb 23 2014


MAPLE

ispoweroftwo := proc(n) local a, t; t := 1; while (n > t) do t := 2*t end do; if (n = t) then a := true else a := false end if; return a; end proc; f:= proc(n) if (not isprime(n)) and (not ispoweroftwo(n)) then return n end if; end proc; seq(f(i), i = 1..150);


MATHEMATICA

max=6!; lst={}; Do[z=n+(n+1); Do[z+=(n+x); If[z>max, Break[]]; AppendTo[lst, z], {x, 2, max}], {n, max}]; Union[lst] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)


CROSSREFS

Cf. A000040, A000079, A066542 (the complement).
Sequence in context: A103092 A104523 A091886 * A036347 A129492 A241913
Adjacent sequences: A111771 A111772 A111773 * A111775 A111776 A111777


KEYWORD

easy,nonn


AUTHOR

Jaap Spies, Aug 15 2005


STATUS

approved



