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A321576 a(n) is the smallest b > 1 such that b^n - (b-1)^n has all divisors d == 1 (mod n).

%I A321576 #23 May 28 2022 12:42:22
%S A321576 2,2,2,3,2,4,2,45,3,6,2,301,2,15,10,121,2,64,2,2101,7,12,2,1900081,6,
%T A321576 27,18,225,2,9241,2,31825,12,52,31,537850405,2,96,26,13568281,2,232,2,
%U A321576 35421,486,24,2,4164776161,7,2101,68,10765,2,145180,1925
%N A321576 a(n) is the smallest b > 1 such that b^n - (b-1)^n has all divisors d == 1 (mod n).
%C A321576 For n > 1, a(n) is the least b > 1 such that b^n - (b-1)^n has all prime divisors p == 1 (mod n).
%C A321576 If n is prime, then a(n) = 2. Conjecture: If n is composite, then a(n) > 2.
%C A321576 From _Kevin P. Thompson_, May 27 2022: (Start)
%C A321576 Sequence continues for n = 56..95 (unconfirmed terms marked with a '?'): 20301625?, 171, 30, 2, ?, 2, 156, 18298, 405825?, 442, 361285?, 2, 8365, 553, 392106?, 2, ?, 2, 75, 4975?, 31351?, 1914, 247339?, 2, ?, 1513?, 42, 2, ?, 391, 87, 406?, ?, 2, ?, 39, ?, 63, 142, 145
%C A321576 a(60) > 1.3831*10^10.
%C A321576 a(72) > 1.34*10^8.
%C A321576 a(80) > 10^8.
%C A321576 a(84) > 2.29*10^8.
%C A321576 a(88) > 10^7.
%C A321576 a(90) > 10^8.
%C A321576 a(92) > 10^6. (End)
%H A321576 FactorDB, <a href="http://factordb.com/index.php?id=1100000002995815526">Status of 20301625^56-20301624^56</a>
%H A321576 Kevin P. Thompson, <a href="/A321576/a321576.txt">Factorizations to support known terms for n = 1..95</a>
%e A321576 a(6) = 4 since b^n - (b-1)^n = 4^6 - 3^6 = 3367 has divisors 1, 7, 13, 37, 91, 259, 481, and 3367, each of which is congruent to 1 (mod 6), and b = 4 is the smallest such number satisfying this requirement.
%t A321576 primes[n_]:=First@# & /@ FactorInteger[n]; bQ[m_, n_]:=AllTrue[primes[m] -1, Divisible[#, n]&] ; a[n_]:=Module[{b=2}, While[!bQ[b^n - (b-1)^n, n], b++]; b]; Array[a, 100] (* _Amiram Eldar_, Nov 13 2018 *)
%o A321576 (PARI) A321576(n)=if(n<4||isprime(n),2,for(b=2,oo,Set(factor(b^n-(b-1)^n)[,1]%n)==[1]&&return(b))) \\ _M. F. Hasler_, Nov 18 2018
%Y A321576 Cf. A298076.
%K A321576 nonn,more
%O A321576 1,1
%A A321576 _Thomas Ordowski_, Nov 13 2018
%E A321576 a(12)-a(23) from _Amiram Eldar_, Nov 13 2018
%E A321576 a(24)-a(55) from _Kevin P. Thompson_, May 27 2022