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

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, 27, 18, 225, 2, 9241, 2, 31825, 12, 52, 31, 537850405, 2, 96, 26, 13568281, 2, 232, 2, 35421, 486, 24, 2, 4164776161, 7, 2101, 68, 10765, 2, 145180, 1925

Offset: 1

Author

Thomas Ordowski, Nov 13 2018

Keywords

• nonn
• more

Comments

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).

If n is prime, then a(n) = 2. Conjecture: If n is composite, then a(n) > 2.

From Kevin P. Thompson, May 27 2022: (Start)

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

a(60) > 1.3831*10^10.

a(72) > 1.34*10^8.

a(80) > 10^8.

a(84) > 2.29*10^8.

a(88) > 10^7.

a(90) > 10^8.

a(92) > 10^6. (End)

Examples

			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.
		

Links

• FactorDB, Status of 20301625^56-20301624^56
• Kevin P. Thompson, Factorizations to support known terms for n = 1..95

Crossrefs

Cf. A298076.

Programs

Mathematica

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 *)

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

Extensions

a(12)-a(23) from Amiram Eldar, Nov 13 2018

a(24)-a(55) from Kevin P. Thompson, May 27 2022