If p is a prime and k is an even integer, 0 < k < p-1, then k is said to be an irregular index for p if p divides the numerator of the k-th Bernoulli number. The index of irregularity of p is the number of irregular indices for p.

The irregular indices for the 9,163,831 primes up to 39*2^22 = 163,577,356 can be found (together with further data explained below) in the file To163.gz

The md5sum of the ungzipped file is a9b0eb7c97a5e9d2163936ea145d70a6

The number of primes of the various indices of irregularity is:

The primes of index 7 are: 3238481, 5216111, 5620861, 9208289, 32012327

The primes p such that p-3 is an irregular index are 16843 and 2124679

A line in the file starts with a prime, and contains clumps of the form k:v,s,t for each irregular index k (v,s,t are explained below). In particular, if P is a regular prime (index of irregularity 0) then the prime is the only thing on the line.

If k is an irregular index for p, L the smallest prime congruent to 1 mod p, m = (L-1)/p, and w = 2^(m/2) mod L, then v := V^m mod L where

V := prod_{a=1}^{(p-1)/2} (w^a-w^{-a})^{a^{p-1-k}} mod L

If V is unequal to 1 for all k for a given p then Vandiver's conjecture is true for p.

To explain the numbers s, t let

S(e) := \sum_{a=1}^{(p-1)/2) a^e mod p^2.

Then s = S(k-1)/p and t = S(p+k-2)/p. (If s or t aren't integral then (p,k) isn't irregular.) If s nonzero mod p, and s is not equal to (-k/2) t mod p, then the numerator of the k-th Bernoulli number isn't divisible by p^2, and the lambda invariant is just equal to the index of irregularity. (This test of the lambda invariant fails for 3 primes in the table, and has to be augmented with a different test.)

Fifteen of the lines contain, in addition to the data above, a (cryptic) comment as to why the line is ``interesting,'' surrounded by semicolons. For instance, the line containing the largest prime of index 7 is

32012327 11016400:175939647,15703970,13838375 12596980:303121954,21688583,20692880 15062986:44038344,28632266,24230253 15329230:364060109,3834460,10374401 19631338:422859016,15465672,6561073 27353368:120022702,24891973,10272561 28255946:22874060,10489245,1763271; Irr = 7;