OFFSET
1,1
COMMENTS
The number k = pq is a weak pseudoprime to prime bases p and q.
Problem: are there infinitely many such numbers?
All the terms are odd squarefree semiprimes.
Semiprimes pq such that p^(p-1) == 1 (mod q) and q^(q-1) == 1 (mod p).
Odd semiprimes pq such that (q-p)^(q-p) == 1 (mod pq).
Semiprimes pq > 6 such that (q-p)^(q-p) == 1 (mod pq).
Odd semiprimes pq pseudoprime to base q-p.
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..2812
EXAMPLE
65 = 5*13 is a term since 5^65 == 5 (mod 65) and 13^65 == 13 (mod 65).
Equivalently: 5^(5-1) == 1 (mod 13) and 13^(13-1) == 1 (mod 5).
Also (13-5)^(5*13-1) == 1 (mod 5*13) or (13-5)^(13-5) == 1 (mod 5*13).
MATHEMATICA
k = 4; lst = {}; NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sp}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega@ sp != 2, If[ sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; fQ[k_] := Block[{fi = First@# & /@ FactorInteger@ k}, PowerMod[#, k, k] & /@ fi == fi]; While[k < 100000, If[ fQ@ k, AppendTo[lst, k]]; k = NextSemiPrime@ k] (* Robert G. Wilson v, Feb 10 2018 *)
PROG
(PARI) lista(nn) = {for (n=1, nn, if (bigomega(n) == 2, if (omega(n) == 2, p = factor(n)[1, 1]; q = factor(n)[2, 1]; , p = factor(n)[1, 1]; q = factor(n)[1, 1]; ); mp = Mod(p, n); mq = Mod(q, n); if ((mp^n == mp) && (mq^n == mq), print1(n, ", ")); ); ); } \\ Michel Marcus, Feb 10 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Ordowski, Feb 10 2018
EXTENSIONS
More terms from Michel Marcus, Feb 10 2018
Edited by Thomas Ordowski, Mar 12 2019
STATUS
approved