Fun with partition function

[Batalov]

Originally Posted by Batalov

Partition function is available in GP/Pari (p(n) = numbpart(n)), but it is slow for large values of n. It is possible to calculate p(n) with the Arb implementation.

...the first (PR)prime value for n>=10¹⁰ is p(10000076282) and has 111391 decimal digits.

To put a symmetrical coda on this thread:
the first (PR)prime value for n>=10¹¹ is p(100000135540) and has 352269 decimal digits.

[paulunderwood]

Originally Posted by Batalov

To put a symmetrical coda on this thread:
the first (PR)prime value for n>=10¹¹ is p(100000135540) and has 352269 decimal digits.

Congrats!

[JM Montolio A]

Some time ago, i find:

E(n,L)=SUM(i=1..i=L, E(n-L,i) )
Number of partitions of n with L numbers.

G(n,L) = G(n-L,L-1) + G(n-L,L).
Number of partitions of n with L different numbers.

Start values E y G :
N E(N,L)
1 1 ,2 1,1 ,3 1,1,1 ,4 1,2,1,1

N G(N,L)
1 1,2 1 ,3 1,1 ,4 1,1 ,5 1,2 ,6 1,2,1
7 1,3,1 ,8 1,3,2

Then:

p(N) = SUM(L=1 .. L=N, E(N,L) ).

p(n) = E(2n,n)

p(N) = G(N + (1+2+...+N), N) = G( (1+2+...+N), N-1)+1

q(N) = SUM G(N,*).

p(100) =4*43*59*89*211.
q(180) =141231780.

Happy to share it here.

JM M