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

1,1

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)

Links

Table of n, a(n) for n=1..55.

FactorDB, Status of 20301625^56-20301624^56

Kevin P. Thompson, Factorizations to support known terms for n = 1..95

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A298076.

Sequence in context: A307408 A233539 A317223 * A304101 A278636 A376497

Adjacent sequences: A321573 A321574 A321575 * A321577 A321578 A321579

Keyword

nonn,more

Author

Thomas Ordowski, Nov 13 2018

Extensions

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

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

Status

approved