Name of equation??

[Peter Gunn]

Hi,

does anyone know the equation f(x) which
calculates the number of possible ways of
calculating x by summing positive integers <=x ???

For instance...

f(1) = 1 [1]
f(2) = 2 [2,1+1]
f(3) = 3 [3,2+1,1+1+1]
f(4) = 5 [4,3+1,2+2,2+1+1,1+1+1+1]
f(5) = 7 [5,4+1,3+2,3+1+1,2+1+1+1,1+1+1+1+1,2+2+1]
f(6) = 11 [6,5+1,4+2,4+1+1,3+3,3+2+1,3+1+1+1,
2+1+1+1+1,1+1+1+1+1+1,2+2+2,2+2+1+1]
... and so on.

thanks,

PG.

[spam...@nil.nil]

Peter Gunn <pete...@NOSPAMhotmail.com> wrote:

> does anyone know the equation f(x) which
> calculates the number of possible ways of
> calculating x by summing positive integers <=x ???

> For instance...

> f(1) = 1 [1]
> f(2) = 2 [2,1+1]

The number of partitions of x ("partitions").

The coefficient of x^n in 1/f(x) where:

f(x)=(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)...
=PRODuct[(1-x^k): for k=1,2,3,4,...]

is the number of partitions of n.

[Peter Gunn]

spam...@nil.nil wrote:

Yes, but I was looking for a simple mechanism :-)
Ive just uncovered the Hardy-Ramanujan approximation...

f(n)=floor(0.5+exp(PI*sqrt(2n/3))/(4n*sqrt(3)))

and apparently the exact sequence was refined by
Rademacher to give...

1 d / e^(PI sqrt(2n/3 - 1/36)) \
f(n)= ----------- --( ------------------------- ) + O(e^(k sqrt(n))
2PI sqrt(2) dn \ sqrt(n - 1/24) /

which is nice, but unfortunately my mathemetical ability
is somewhat nonexistant, so could someone please help me turn
this into a 'C style' equation like the Hardy-Ramanujan
approximation??

Also, does anyone know a quick method of determining
n given f(n) without iteration or recursion, a way to
calculate f(xy) given f(x) and f(y), or a way to
calculate f(xy) given x and f(y)??

Thanks,

PG.

[Josephine Michelle Draus and Lorraine Lee]

Peter Gunn wrote in message <371B4E5...@NOSPAMhotmail.com>...

>spam...@nil.nil wrote:
>
>> Peter Gunn <pete...@NOSPAMhotmail.com> wrote:
>>
>> > does anyone know the equation f(x) which
>> > calculates the number of possible ways of
>> > calculating x by summing positive integers <=x ???

/0 (k>n)
p(k,n)={ 1 (k=n)
\p(k+1,n)+p(k,n-k) (k<n)

Number of partitions of n is p(1,n).

p(k,n) is number of partitions of n,
allowing only addends >=k.

[Kurt Foster]

In <371B4E5...@NOSPAMhotmail.com>, Peter Gunn said:
: Yes, but I was looking for a simple mechanism :-)

: Ive just uncovered the Hardy-Ramanujan approximation...

Just out of curiosity, in which of the sources to which I referred you
did you "uncover" Hardy-Ramanujan, or Rademacher's formulation?

[dr_a...@my-dejanews.com]

In article <7fg356$8v9$1...@news1.rmi.net>,

Isn't the Hardy Ramanujan identities in Hardy's book :
Hardy and Wright : An introduction to number theory.
(i guess not, they found the results after that).
Barring that, the result is quoted, i am sure, in
Tom Apostol : Analytic Number Theory.

Cheers.

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[Peter Gunn]

Kurt Foster wrote:

> In <371B4E5...@NOSPAMhotmail.com>, Peter Gunn said:
> : Yes, but I was looking for a simple mechanism :-)
> : Ive just uncovered the Hardy-Ramanujan approximation...
>
> Just out of curiosity, in which of the sources to which I referred you
> did you "uncover" Hardy-Ramanujan, or Rademacher's formulation?

Dont want to sound like a smart arse (coz Im not :-) but I
found this before I read your email...

http://www.seanet.com/~ksbrown/kmath266.htm

Its worth noting that I listed the Ramanujan and
Hardy-Ramanujan approximations and not the
Rademacher one, I think :-)

Thanks for the help anyway, I'll try and find an online
version of the texts you mentioned.

If you find the Rademacher info please email :-)

ttfn

PG.

[Steve Monson]

Peter Gunn wrote:
>
> Hi,

>
> does anyone know the equation f(x) which
> calculates the number of possible ways of
> calculating x by summing positive integers <=x ???
>

> For instance...
>
> f(1) = 1 [1]
> f(2) = 2 [2,1+1]

> f(3) = 3 [3,2+1,1+1+1]
> f(4) = 5 [4,3+1,2+2,2+1+1,1+1+1+1]
> f(5) = 7 [5,4+1,3+2,3+1+1,2+1+1+1,1+1+1+1+1,2+2+1]
> f(6) = 11 [6,5+1,4+2,4+1+1,3+3,3+2+1,3+1+1+1,
> 2+1+1+1+1,1+1+1+1+1+1,2+2+2,2+2+1+1]

Take a look at

http://www.seanet.com/~ksbrown/kmath266.htm

for an asymptotic formula for the number of partitions for N. It also
gives a reference for further study. I'm sure Sloane's Handbook of
Integer Sequences will also give you some further values.

http://www.seanet.com/~ksbrown/kmath383.htm

also discusses this a bit.

Steve Monson

[Kurt Foster]

Rademacher's formula is given in "Handbook of Mathematical Functions"
by Abramowitz and Stegun. See Tom Apostol's books, "Introduction to
Analytic Number Theory" and the sequel (Modular Functions and Number
Theory?) for the derivations. I think it's also in a book by Rademacher
himself. And no doubt innumerable other sources.
FWIW, "Handbook" also lists the values of p(n) [unrestricted partitions]
and q(n) [partitions into distinct parts] for n = 1 to n = 500. It gives

p(500) = 2300165032574323995027
q(500) = 732986521245024