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 A295997 Least composite k such that d^k == d (mod k) for every divisor d of n. 3
 4, 341, 6, 341, 4, 561, 6, 341, 6, 561, 10, 561, 4, 561, 561, 341, 4, 561, 6, 561, 6, 561, 22, 561, 4, 561, 6, 561, 4, 561, 6, 341, 561, 561, 561, 561, 4, 561, 6, 561, 4, 561, 6, 561, 561, 341, 46, 561, 6, 561, 91, 561, 4, 561, 10, 561, 6, 341, 15, 561, 4, 341 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(n) is the smallest weak pseudoprime k to all natural bases d|n. For n > 1, a(n) is the smallest composite k such that p^k == p (mod k) for every prime p dividing n; so a(n) is the smallest weak pseudoprime k to all prime bases p|n (thus it is enough to check this congruence only for all prime divisors p of n, see the second program in Pari). For n > 1, a(n) = 4 iff n has all prime divisors p == 1 (mod 4). The sequence is bounded, namely 4 <= a(n) <= 561, see A002997. All members of A108574 appear in the sequence. The last to appear is 538 = a(8110351). - Robert Israel, Feb 15 2018 Conjecture: all distinct terms of the sequence are A108574. - Robert Israel and Thomas Ordowski, Feb 16 2018. The conjecture is true and can be established computationally, like in Conway-Guy-Schneeberger-Sloane (1997) paper. - Max Alekseyev, Feb 27 2018 Note that a(n) >= A000790(n). - Thomas Ordowski, Feb 16 2018 The sequence is not eventually periodic: e.g., any arithmetic progression contains infinitely many terms divisible by a prime == 3 (mod 4), and thus with a(n) > 4, while on the other hand there are infinitely many terms with a(n) = 4. - Robert Israel, Feb 16 2018 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 J. H. Conway, R. K. Guy, W. A. Schneeberger, and N. J. A. Sloane, The Primary Pretenders, Acta Arith. 78 (1997), 307-313. FORMULA a(n) = a(rad(n)), where rad(n) = A007947(n). For prime p, a(p) = A000790(p). - Max Alekseyev, Feb 27 2018 MAPLE f := n -> g(map(t -> t[1], ifactors(n)[2])): g:= proc (P) local k; option remember;   for k from 4 do     if not isprime(k) and andmap(p -> (p &^ k - p mod k = 0), P)     then return k     end if   end do end proc: map(f, [\$1..100]); # Robert Israel, Feb 14 2018 MATHEMATICA With[{c = Table[FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1], {n, 561}]}, Table[With[{d = Divisors@ n}, SelectFirst[c, Function[k, AllTrue[d, PowerMod[#, k, k] == Mod[#, k] &]]]], {n, 62}]] (* Michael De Vlieger, Feb 17 2018, after Robert G. Wilson v at A066277 *) PROG (PARI) a(n) = forcomposite(k=1, , my (ok=1); fordiv (n, d, if (Mod(d, k)!=Mod(d, k)^k, ok=0; break)); if (ok, return (k))); \\ Rémy Sigrist, Feb 14 2018 (PARI) a(n)=my(f=factor(n)[, 1], p); forcomposite(k=4, 561, for(i=1, #f, p=f[i]; if(Mod(p, k)^k!=p, next(2))); return(k)); \\ Charles R Greathouse IV, Feb 14 2018 CROSSREFS Cf. A000790, A002808, A002997, A007947, A108574. Sequence in context: A214161 A265868 A239293 * A090086 A007535 A000783 Adjacent sequences:  A295994 A295995 A295996 * A295998 A295999 A296000 KEYWORD nonn AUTHOR Thomas Ordowski, Feb 14 2018 EXTENSIONS More terms from Rémy Sigrist, Feb 14 2018 STATUS approved

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Last modified April 16 17:01 EDT 2021. Contains 343050 sequences. (Running on oeis4.)