Math Is Fun Forum

pellerinb

Member

Registered: 2012-12-26

Posts: 43

Smarandache Reverse Power Summation (SRPS) numbers

I read an article about SRPS numbers and thought it would be neat to explore this sequence a little more. The article is available on the Free Online Library titled as above by Jason Earls.

I notice similarities between the set of SPRS numbers (1, 3, 8, 22, 65, ...) and the set of natural numbers (1, 2, 3, 4, 5, ...). In either set,

2/4 = 1/2 of the numbers are divisible by 2.

6/18 = 1/3 of the numbers are divisible by 3.

but 3/8 of the SPRS numbers are divisible by 4 (as opposed to 2/8 = 1/4 of the natural numbers)

yet 20/100 = 1/5 of the numbers are divisible by 5.

6/36 = 1/6 of the numbers are divisible by 6.

42/294 = 1/7 of the numbers are divisible by 7.

but 3/16 of the numbers are divisible by 8.

6/54 = 1/9 of the numbers are divisible by 9.

but 3/32 of the numbers are divisible by 16.

I conjecture that 3/2^(n+1) of the SPRS numbers are divisible by 2^n for all n > 1 and 1/m of the SPRS numbers are divisible by all other numbers m not represented by 2^n.

Perhaps you find this interesting too. Thanks in advance for your comments, corrections and the like!

Sincerely,
Brian Pellerin, MSc Dalhousie

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Prime numbers have got to be the neatest things; they are like atoms. Composites are two or more primes held together by multiplication.
In biology, we use math like we know what we are talking about. Sad isn't it.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Smarandache Reverse Power Summation (SRPS) numbers

Hi pellerinb;

I did not notice a download link for article. Are you aware of one?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

pellerinb

Member

Registered: 2012-12-26

Posts: 43

Re: Smarandache Reverse Power Summation (SRPS) numbers

I cannot post a download link because I am not an "established" member yet. However, google "smarandache reverse power summation numbers the free library" and it should be the first hit.

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Prime numbers have got to be the neatest things; they are like atoms. Composites are two or more primes held together by multiplication.
In biology, we use math like we know what we are talking about. Sad isn't it.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Smarandache Reverse Power Summation (SRPS) numbers

Hi;

Okay, I got the paper.

For your assertion, can I read 20 / 100 to imply that 20 out of the first 100 numbers are divisible by 5  and likewise 6/36 that 6 out of the first 36 numbers are divisible by 6?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

pellerinb

Member

Registered: 2012-12-26

Posts: 43

Re: Smarandache Reverse Power Summation (SRPS) numbers

Absolutely. 20 / 100 implies that 20 out of the first 100 numbers are divisible by 5, etc. It's apparently a repeating pattern too, so the next 100 numbers are divisible by 5 as were the first.  Entering "[Table[sum from k = 1 to T (T-k+1)^k, {T,1,200}]] mod 5" in Wolfram|Alpha shows the residues modulo 5 for the first 200 SRPS numbers.

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Prime numbers have got to be the neatest things; they are like atoms. Composites are two or more primes held together by multiplication.
In biology, we use math like we know what we are talking about. Sad isn't it.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Smarandache Reverse Power Summation (SRPS) numbers

Hi pellerinb;

Okay, thank you. It is something to play around with.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.