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A208728 Composite numbers n such that b^(n+1) == 1 (mod n) for every b coprime to n. 46
15, 35, 255, 455, 1295, 2703, 4355, 6479, 9215, 10439, 11951, 16211, 23435, 27839, 44099, 47519, 47879, 62567, 63167, 65535, 93023, 94535, 104195, 120959, 131327, 133055, 141155, 142883, 157079, 170819, 196811, 207935, 260831, 283679, 430199, 560735, 576719 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

GCD(b,n)=1 and b^(n+1) == 1 (mod n).

The sequence lists the squarefree composite numbers n such that every prime divisor p of n satisfies (p-1)|(n+1) (similar to Korselt's criterion).

The sequence can be considered as an extension of k-Knödel numbers to k negative, in this case equal to -1.

Numbers n > 3 such that b^(n+2) == b (mod n) for every integer b. Also, numbers n > 3 such that A002322(n) divides n+1. Are there infinitely many such numbers? It seems that such numbers n > 35 have at least three prime factors. - Thomas Ordowski, Jun 25 2017

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Carmichael Number

Eric Weisstein's World of Mathematics, Korselt's Criterion

Eric Weisstein's World of Mathematics, Knödel Numbers

EXAMPLE

6479 is part of the sequence because its prime factors are 11, 19 and 31: (6479+1)/(11-1)=648, (6479+1)/(19-1)=360 and (6479+1)/(31-1)=216.

MAPLE

with(numtheory); P:=proc(n) local d, ok, p;

if issqrfree(n) then p:=factorset(n); ok:=1;

for d from 1 to nops(p) do if frac((n+1)/(p[d]-1))>0 then ok:=0;

break; fi; od; if ok=1 then n; fi; fi; end: seq(P(i), i=5..576719);

MATHEMATICA

Select[Range[2, 576719], SquareFreeQ[#] && ! PrimeQ[#] && Union[Mod[# + 1, Transpose[FactorInteger[#]][[1]] - 1]] == {0} &] (* T. D. Noe, Mar 05 2012 *)

PROG

(PARI) is(n)=if(isprime(n)||!issquarefree(n)||n<3, return(0)); my(f=factor(n)[, 1]); for(i=1, #f, if((n+1)%(f[i]-1), return(0))); 1 \\ Charles R Greathouse IV, Mar 05 2012

CROSSREFS

Cf. A002322, A002997, A006972, A033553, A050990, A050992, A050993, A208154-A208158.

Sequence in context: A037074 A107423 A027442 * A219689 A074891 A328213

Adjacent sequences:  A208725 A208726 A208727 * A208729 A208730 A208731

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Mar 01 2012

EXTENSIONS

Definition corrected by Thomas Ordowski, Jun 25 2017

STATUS

approved

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Last modified June 24 18:43 EDT 2021. Contains 345419 sequences. (Running on oeis4.)