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May 31st, 2013, 01:28 AM   #1
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Powers expressed as sum of consecutive numbers



.

Option 1) If "n"=couple










Demonstration:




Option 2) If "n"=odd










Demonstration:




Examples:
1)






2)






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May 31st, 2013, 02:52 AM   #2
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Re: Powers expressed as sum of consecutive numbers

Indeed, not just powers, but any composite number can be expressed as the sum of consecutive numbers, eg 35 = 5*7 = 5+6+7+8+9 = 2+3+4+5+6+7+8

Therefore, any composite can be expressed as the difference between triangular numbers, eg 35 45-10 or 36-1
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May 31st, 2013, 03:46 AM   #3
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Re: Powers expressed as sum of consecutive numbers

Primes too. for example: 5 = 15 - 10 and 17 = 153 - 136.
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May 31st, 2013, 05:36 AM   #4
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Re: Powers expressed as sum of consecutive numbers

Nice problem! See A001227 in the OEIS: number of ways to write n as difference of two triangular numbers; number of odd divisors of n.
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May 31st, 2013, 02:44 PM   #5
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Re: Powers expressed as sum of consecutive numbers

Quote:
Originally Posted by Hoempa
Primes too. for example: 5 = 15 - 10 and 17 = 153 - 136.
Indeed. Primes can be the difference either of consecutive triangles, since any number at all can obviously be so expressed:

66 - 55 = 11, since 1+2+3+4+5+6+7+8+9+10+11 - 1+2+3+4+5+6+7+8+9+10 = 11

Primes can also be the difference between triangles spaced two apart, as every odd number can be so expressed:

1+5+3+4+5+6 - 1+5+3+4 = 5+6 = 11

But that's it.

Composites have other possibilities.
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May 31st, 2013, 05:46 PM   #6
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Re: Powers expressed as sum of consecutive numbers

Quote:
Originally Posted by mente oscura


Option 2) If "n"=odd







Curious application:







Demonstration:











Example:


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June 1st, 2013, 02:11 AM   #7
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Re: Powers expressed as sum of consecutive numbers

Quote:
Originally Posted by johnr
Composites have other possibilities
Not all of them. What about n = 4? See the sequence CRGreathouse gave.
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June 1st, 2013, 02:48 AM   #8
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Re: Powers expressed as sum of consecutive numbers

Quote:
Originally Posted by Hoempa
Quote:
Originally Posted by johnr
Composites have other possibilities
Not all of them. What about n = 4? See the sequence CRGreathouse gave.
Oh, sorry. Yes, evens are a different story. I should have specified that I was, as usual, talking about odd composites.

Yes, powers of 2 are even more restricted than odd primes in how they can be expressed as the difference between triangular numbers.

I never thought about this stuff simply in terms of how many odd factors a number has and find CRG's link quite fascinating!
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