# [seqfan] n=3^k3+5^k5+7^k7 (Was: a simple Partitions question)

zak seidov zakseidov at yahoo.com
Sun Mar 21 05:33:37 CET 2010

```Dear fellow seqfans,

Sorry for loosely related (off-)topic...

Representation of n as the sum of positive powers of 3, 5 and 7:
n = 3^k3 + 5^k5 + 7^k7, (with positive integers k3, k5, k7).

Seven cases with two representations:
n,            {k3,k5,k7}
n=135, {1,3,1},{4,1,2}
n=255, {4,3,2},{5,1,1}
n=375, {3,1,3},{5,3,1}
n=2535,{2,3,4},{7,1,3}
n=3135, {1,5,1},{6,1,4}
n=3155, {6,2,4},{7,4,3}
n=3255, {4,5,2},{6,3,4}
E.g,
135=3+125+7=81+5+49
255=81+125+49=243+5+7
375=27+5+343=243+125+7.

Are these only cases with more than one representation(s)?
What about other bases different from {3,5,7}?

Thanks,
Zak

----- Original Message ----
From: Don Reble <djr at nk.ca>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Sat, March 20, 2010 10:13:41 PM
Subject: [seqfan] Re: a simple Partitions question

> strict partitions of n into positive powers of 3, 5 and 7,
> ... the count of such partitions, say P357(n), ... shows
> surprising symmetry:
> P357(n) = P357(2270-n) for n>84

You mean 2271-n.

It's symmetric because
2271 = 3^1+3^2+3^3+3^4+3^5+3^6 + 5^1+5^2+5^3+5^4 + 7^1+7^2+7^3
That is, all powers less than 2187 (=3^7) contribute.
So for any partition of n, with (2271-2187) < n < 2187,
there's a complimentary partition for 2271-n.

--
Don Reble  djr at nk.ca

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