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Replies: 3   Last Post: Mar 7, 2005 3:26 PM

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 Kevin Karplus Posts: 190 Registered: 12/6/04
Posted: Jan 2, 2005 9:44 AM

In article <vmret05okibpmp42l1apqil3530agrmqa1@4ax.com>, EJC wrote:
> Which positive # less than a 100 cannot be expressed as a sum of a
> series of consecutive pos. integers?

1) Technically, a "series" is an infinite sum, and any series of
positive integers will not converge.

2) If you just meant "some number of consecutive positive integers", then
all positive integers can be so represented, as a sum of precisely
one positive integer (itself).

You probably meant something different, like "what positive number less
than 100 cannot be expressed as a sum of 2 or more consecutive
positive integers?" It is very important in math to be precise in

Or perhpas you meant "what positive numbers ...?" The difference
between the singular and the plural is important---do you need all the
numbers that have that property, or just one example?

You can answer the question fairly easily by considering that
numbers congurent to 1 mod 2 (2n+1) can be represented as n+(n+1),
numbers congruent to 0 mod 3 (3n) can be represented as (n-1)+n+(n+1)
numbers congruent to 2 mod 4 (4n+2) as (n-1)+n+(n+1)+(n+2),
numbers congruent to 0 mod 5 (5n) as (n-2)+(n-1)+n+(n+1)+(n+2)
numbers congruent to 3 mod 6 (6n+3) as (n-2)+(n-1)+n+(n+1)+(n+2)+(n+3),
numbers congruent to 0 mod 7 (7n)
as (n-3)+(n-2)+(n-1)+n+(n+1)+(n+2)+(n+3)
numbers congruent to 4 mod 8 (8n+4)
as (n-3)+(n-2)+(n-1)+n+(n+1)+(n+2)+(n+3)+(n+4)
...

If you continue in this way you will cover all the positive integers,
except those whose values are so small that the first terms of the sum
are not positive integers. (For 2n+1, need n>0, for 3n, need (n-1)>0,
for 4n+2, need (n-1)>0, ...)

For a limit as low as 100, it is probably easiest to make a sieve,
crossing out the numbers that CAN be represented as a sum of 2, 3, 4,
... consecutive integers.

------------------------------------------------------------
Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus
Professor of Biomolecular Engineering, University of California, Santa Cruz
(Senior member, IEEE) (Board of Directors, ISCB starting Jan 2005)
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Affiliations for identification only.

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Date Subject Author
1/2/05 Kevin Karplus
3/7/05 Joshua Zucker
1/14/05 Algebryonic