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A208728
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Composite numbers n such that b^(n+1) == 1 (mod n) for every b coprime to n.
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46
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MAPLE
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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);
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MATHEMATICA
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Select[Range[2, 576719], SquareFreeQ[#] && ! PrimeQ[#] && Union[Mod[# + 1, Transpose[FactorInteger[#]][[1]] - 1]] == {0} &] (* T. D. Noe, Mar 05 2012 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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