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A194231 Numbers n such that at least one of n and n+2 is composite, while for every b coprime to n*(n+2), b^(n-1)==1 (mod n) and b^(n+1)==1 (mod n+2). 1
561, 1103, 2465, 2819, 6599, 29339, 41039, 52631, 62743, 172079, 188459, 278543, 340559, 488879, 656599, 656601, 670031, 1033667, 1909001, 2100899, 3146219, 5048999, 6049679, 8719307, 10024559, 10402559, 10877579, 11119103, 12261059, 14913989, 15247619 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

These might be called "Carmichael pseudo-twin-primes".

LINKS

Table of n, a(n) for n=1..31.

FORMULA

For every b coprime to a(n)*(a(n)+2), 2*b^(a(n)+1)==(b^2-1)*(a(n)+2) (mod a(n)*(a(n)+2)). Conversely (Max Alekseyev), if for every b coprime to N*(N+2), 2*b^(N+1)==(b^2-1)*(N+2) (mod N*(N+2)), then N is in the sequence. - Vladimir Shevelev, Oct 14 2011

MAPLE

with(numtheory):

ic:= proc(n) local p;

       if not issqrfree(n) then false

     else for p in factorset(n) do

            if irem (n-1, p-1)<>0 then return false fi

          od; true

       fi

     end:

a:= proc(n) option remember; local k;

      for k from 2 +`if`(n=1, 1, a(n-1)) by 2 while

        isprime(k) and isprime(k+2) or not (ic(k) and ic(k+2))

      do od; k

    end:

seq(a(n), n=1..10);  # Alois P. Heinz, Oct 12 2011

MATHEMATICA

terms = 31; bMax = 20(* sufficient for 31 terms *); coprimes[n_] := Select[ Range[bMax], CoprimeQ[#, n]&]; Reap[For[n = m = 1, m <= terms, n += 2, If[CompositeQ[n] || CompositeQ[n+2], If[AllTrue[coprimes[n(n+2)], PowerMod[#, n-1, n] == 1 && PowerMod[#, n+1, n+2] == 1&], Print["a(", m, ") = ", n]; Sow[n]; m++]]]][[2, 1]] (* Jean-François Alcover, Mar 28 2017 *)

CROSSREFS

Cf. A002997, A001567, A141232.

Sequence in context: A085999 A224695 A137198 * A141705 A135721 A290486

Adjacent sequences:  A194228 A194229 A194230 * A194232 A194233 A194234

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Oct 12 2011

STATUS

approved

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Last modified November 18 10:18 EST 2017. Contains 294887 sequences.