# [seqfan] On smallest PSP(n) of the repunit form R_n(k) = (n^k-1)/(n-1)

Tomasz Ordowski tomaszordowski at gmail.com
Wed May 2 05:56:59 CEST 2018

```Dear SeqFan,

I have something interesting:

Let a(n) be the smallest k such that that R_n(k) = (n^k-1)/(n-1) is a
psp(n) for n > 1.

? a(n) is the smallest k > 1 such that R_n(k) == 1 (mod k), i.e., n^k == n
(mod (n-1)k).

Data: 11, 5, 4, 5, 6, 7, 2, ... Please more terms.

b(n) = R_n(a(n)) = (n^a(n)-1)/(n-1) : 2047, 121, 85, 781, 9331, 137257, 9,
...
________________________
Steuerwald's theorem (1948): If k is a weak psp(b) and gcd(k,b-1)=1, then
R_b(k) = (b^k-1)/(b-1) is a psp(b).
Denote: The weak psp(b) is a composite k such that b^k == b (mod k). The
psp(b) is a composite k such that b^(k-1) == 1 (mod k).
It should be noted that if p is a prime that does not divide b-1 and R_b(p)
= (b^p-1)/(b-1) is composite, then R_b(p) is a strong psp(b).
http://mathworld.wolfram.com/StrongPseudoprime.html
https://en.wikipedia.org/wiki/Strong_pseudoprime

Sincerely,

Thomas

P.S. Similar sequences:

Let a(n) be the smallest prime p that does not divide n-1 such that R_n(p)
= (n^p-1)/(n-1) is composite for n > 1.

? a(n) is the smallest prime p such that R_n(p) == 1 (mod p), i.e., n^p ==
n (mod (n-1)p).

Data: 11, 5, 5, 5, 11, 7, 2, ... This sequence is also not in the OEIS.

b(n) = R_n(a(n)) = (n^a(n)-1)/(n-1) : 2047, 121, 341, 781, 72559411,
137257, 9, ...

Note that b(n) is a strong psp(n), but not necessarily the smallest
spsp(n).

Cf. https://oeis.org/A298756