Some interesting sequences (Partitions)

[Olivier Gerard]

At 21:32 +0100 98.12.16, David W. Wilson wrote:
>  Here are some interesting sequences of partition numbers mod 11.  No other
> number seems to give such interesting results relative the partitions.
> 

>> Cut list of partitions (11n+k) mod 11 <<

It seems fit to recall some congruence properties of partition numbers,
the most important of which have been discovered by Ramanujan:

p(5n+4)       = 0 mod 5
p(7n+5)       = 0 mod 7
p(11n+6)      = 0 mod 11
p(25n-1)      = 0 mod 25
p(35n+19)     = 0 mod 35
p(49n-2)      = 0 mod 49
p(55n+39)     = 0 mod 55
p(77n+61)     = 0 mod 77
p(121n+116)   = 0 mod 121
p(125n+99)    = 0 mod 125

Ramanujan proved part of them and a few of those he conjectured were
false. G. Watson then A.O.L. Atkin proved the final versions (proofs
are quite long and difficult and involves the powertools of 
Analyic Number Theory, so I won't give even hints...)

So, one has:

p(k*delta + off) = 0 (mod deltabis)

where delta is a number greater than 1 of the form  5^a*7^b*11^c
(a,b,c are integers)

deltabis is delta with the exponent b modified to [(b+2)/2]

off is the initial offset. The condition to compute it is 24 off = 1 (mod delta).

I have produced the corresponding sequences for the EIS : 

(Beware : A-numbers of the 3 new sequences listed here are temporary)

A057110 gives the moduli for which the hypothesis of the theorem are satisfied
A057111 gives the corresponding modified moduli
A057112 gives the corresponding offset <<-- The most interesting sequence



I guess David will be able to produce from that more sequences of the
kind he already sent.


Olivier


%I A057110 
%S A057110 5,7,11,25,35,49,55,77,121,125,175,245,275,343,385,539,605,625,847,
%T A057110 875,1225,1331,1375,1715,1925,2401,2695,3025,3125,3773,4235,4375,5929,
%U A057110 6125,6655,6875,8575,9317,9625,12005,13475,14641,15125,15625,16807
%N A057110 Numbers whose prime factors are in {5, 7, 11}.
%R A057110 Andr76 160.
%D A057110 G.E. Andrews. The Theory of Partitions. p159-161
%Y A057110 Cf. A057111,A057112.
%A A057110 Olivier Gerard (ogerard at ext.jussieu.fr)
%O A057110 0,1
%K A057110 nonn,easy,part

%I A057111 
%S A057111 5,7,11,25,35,49,55,77,121,125,175,245,275,49,385,539,605,625,847,875,
%T A057111 1225,1331,1375,245,1925,343,2695,3025,3125,539,4235,4375,5929,6125,
%U A057111 6655,6875,1225,9317,9625,1715,13475,14641,15125,15625,343,2695,21175
%N A057111 Transformation of A057110:  5^a*7^b*11^c ->  5^a*7^[(b+2)/2]*11^c.
%R A057111 Andr76 160.
%D A057111 G.E. Andrews. The Theory of Partitions. p159-161
%Y A057111 Cf. A057110,A057112.
%A A057111 Olivier Gerard (ogerard at ext.jussieu.fr)
%O A057111 0,1
%K A057111 nonn,easy,part
%C A057111 [x] means integer part of x.

%I A057112 
%S A057112 4,5,6,24,19,47,39,61,116,99,124,194,149,243,369,292,479,599,600,474,
%T A057112 1174,721,974,929,1524,2301,1909,2899,2474,2987,2294,3099,5682,4849,
%U A057112 4714,3724,6074,7376,9224,9504,7299,14031,11974,14974,11905,18079
%N A057112 Ramanujan-Atkin Partition Congruence theorem. For every integer k, p(k*b(n) + a(n)) = 0 (mod c(n)).
%R A057112 Andr76 160.
%D A057112 G.E. Andrews. The Theory of Partitions. p159-161
%Y A057112 Cf. A000041,A057110,A057111.
%A A057112 Olivier Gerard (ogerard at ext.jussieu.fr)
%O A057112 0,1
%K A057112 nonn,easy,part,nice
%C A057112 For each n, a(n) is a term of this sequence, b(n) and c(n) are the corresponding term of A057110 and A057111 and p is the partition function.