My Math Forum Powers expressed as sum of consecutive numbers

 Number Theory Number Theory Math Forum

 May 31st, 2013, 01:28 AM #1 Senior Member     Joined: May 2013 From: España Posts: 151 Thanks: 4 Powers expressed as sum of consecutive numbers $For : a, n, k\in{N}$ $I define : a_1, a_2,..., a_k\in{Q}/a_{i+1}=a_i+1/a^n=a_1+a_2+...+a_k$. Option 1) If "n"=couple $a_1=\dfrac{a^{\frac{n}{2}}+1}{2}$ $a_k=\dfrac{3a^{\frac{n}{2}}-1}{2}$ $k=a^{\frac{n}{2}$ Demonstration: $\sum_{i=1}^k{a_i}=a_1+a_2+...+a_k=\dfrac{\dfrac{a^ {\frac{n}{2}}+1+3a^{\frac{n}{2}}-1}{2}}{2}a^{\frac{n}{2}=a^n$ Option 2) If "n"=odd $a_1=\dfrac{a^{\frac{n+1}{2}}-2a^{\frac{n-1}{2}}+1}{2}$ $a_k=\dfrac{a^{\frac{n+1}{2}}+2a^{\frac{n-1}{2}}-1}{2}$ $k=2a^{\frac{n-1}{2}}$ Demonstration: $\sum_{i=1}^k{a_i}=a_1+a_2+...+a_k=\dfrac{\dfrac{a^ {\frac{n+1}{2}}-2a^{\frac{n-1}{2}}+1+a^{\frac{n+1}{2}}+2a^{\frac{n-1}{2}}-1}{2}}{2}2a^{\frac{n-1}{2}}=a^n$ Examples: 1) $5^4=13+14+15+...+36+37$ $a_1=13$ $a_k=37$ $k=25$ $5^4=\dfrac{13+37}{2}25=625$ 2)$4^7=64,5+65,5+66,5+...+190,5+191,5$ $a_1=64,5$ $a_k=191,5$ $k=128$ $4^7=\dfrac{64,5+191,5}{2}128=16384$ Regards.
 May 31st, 2013, 02:52 AM #2 Math Team   Joined: Apr 2012 Posts: 1,579 Thanks: 22 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
 May 31st, 2013, 03:46 AM #3 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Powers expressed as sum of consecutive numbers Primes too. for example: 5 = 15 - 10 and 17 = 153 - 136.
 May 31st, 2013, 05:36 AM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 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.
May 31st, 2013, 02:44 PM   #5
Math Team

Joined: Apr 2012

Posts: 1,579
Thanks: 22

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.

May 31st, 2013, 05:46 PM   #6
Senior Member

Joined: May 2013
From: España

Posts: 151
Thanks: 4

Re: Powers expressed as sum of consecutive numbers

Quote:
 Originally Posted by mente oscura $For : a, n, k\in{N}$ Option 2) If "n"=odd $a_1=\dfrac{a^{\frac{n+1}{2}}-2a^{\frac{n-1}{2}}+1}{2}$ $a_k=\dfrac{a^{\frac{n+1}{2}}+2a^{\frac{n-1}{2}}-1}{2}$ $k=2a^{\frac{n-1}{2}}$
Curious application:

$S=1^3+3^3+5^3+...+T^3=?$

$a_{kT}=\dfrac{T^2+2T-1}{2}$

$S=\dfrac{1+\dfrac{T^2+2T-1}{2}}{2}\dfrac{T^2+2T-1}{2}=\dfrac{[T(T+2)]^2-1}{8}$

Demonstration:

$For: a\in{N}, n=3/a^3$

$a_1=\dfrac{a^2-2a+1}{2}$

$a_k=\dfrac{a^2+2a-1}{2}$

$For: (a+2), n=3/(a+2)^3$

$a'_1=\dfrac{(a+2)^2-2(a+2)+1}{2}=\dfrac{a^2+4a+4-2a-4+1}{2}=\dfrac{a^2+2a+1}{2}=a_k+1$

Example:
$S=1^3+3^3+5^3+...+37^3=\dfrac{(37*39)^2-1}{8}=260281$

Regards.

June 1st, 2013, 02:11 AM   #7
Math Team

Joined: Apr 2010

Posts: 2,780
Thanks: 361

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.

June 1st, 2013, 02:48 AM   #8
Math Team

Joined: Apr 2012

Posts: 1,579
Thanks: 22

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!

 Tags consecutive, expressed, numbers, powers, sum

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post johnr Number Theory 5 March 5th, 2014 11:03 AM Dacu Number Theory 3 May 31st, 2013 09:06 PM daigo Algebra 1 May 18th, 2012 02:59 PM coax Number Theory 1 July 24th, 2009 05:36 AM natus zeri Math Events 2 December 29th, 2007 01:52 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top