login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A321576 a(n) is the smallest b > 1 such that b^n - (b-1)^n has all divisors d == 1 (mod n). 1
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 (list; graph; refs; listen; history; text; internal format)
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 A126336

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 6 03:14 EDT 2022. Contains 357261 sequences. (Running on oeis4.)