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 A193314 The smallest k such that the product k*(k+1) is divisible by the first n primes and no others. 2
 1, 2, 5, 14, 384, 1715, 714, 633555 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(9)-a(21) do not exist.  It seems unlikely that a(n) exists for larger n. [Charles R Greathouse IV, Aug 18 2011] If a term beyond a(8) exists, it is larger than 2.29*10^25. - Giovanni Resta, Nov 30 2019 LINKS Carlos Rivera, The prime puzzles & problems connection, Puzzle 358. Ruth-Aaron pairs revisited. EXAMPLE n  smallest k   k*(k+1) prime factorization 1  1            2 2  2            2*3 3  5            2*3*5 4  14           2*3*5*7 5  384          2^7*3*5*7*11 6  1715         2^2*3*7^3*11*13 7  714          2*3*5*7*11*13*17 8  633555       2^2*3^3*5*7*11^3*13*17*19^2 MATHEMATICA f[n_] := Block[{k = 1, p = Fold[ Times, 1, Prime@ Range@ n], tst = Prime@ Range@ n}, While[ First@ Transpose@ FactorInteger[ k*p]!=tst || IntegerQ@ Sqrt[ 4k*p+1], k++]; Floor@ Sqrt[k*p]]; Array[f, 8] (* the search for a(9), I also used *) lst = {}; p = Prime@ Range@ 9; Do[ q = {a, b, c, d, e, f, g, h, i}; If[ IntegerQ[ Sqrt[4Times @@ (p^q) + 1]], r = Floor@ Sqrt@ Times @@ (p^q); Print@ r; AppendTo[lst, r]], {i, 9}, {h, 9}, {g, 9}, {f, 10}, {e, 11}, {d, 14}, {c, 16}, {b, 24}, {a, 8}] PROG (PARI) a(n)={   my(v=[Mod(0, 1)], u, P=1, t, g, k);   forprime(p=2, prime(n),     P*=p;     u=List();     for(i=1, #v,       listput(u, chinese(v[i], Mod(-1, p)));       listput(u, chinese(v[i], Mod(0, p)))     );     v=0; v=Vec(u)   );   v=vecsort(lift(v));   while(1,     for(i=1, #v,       t=(v[i]+k)*(v[i]+k+1)/P;       if(!t, next);       while((g=gcd(P, t))>1, t/=g);         if (t==1, return(v[i]+k))     );     k += P   ) }; \\ Charles R Greathouse IV, Aug 18 2011 (Haskell) a193314 n = head [k | k <- [1..], let kk' = a002378 k,                       mod kk' (a002110 n) == 0, a006530 kk' == a000040 n] -- Reinhard Zumkeller, Jun 14 2015 CROSSREFS Cf. A006145, A039945. Cf. A002110, A002378, a006530, A118478. Sequence in context: A081483 A118478 A179675 * A270351 A240435 A146116 Adjacent sequences:  A193311 A193312 A193313 * A193315 A193316 A193317 KEYWORD nonn AUTHOR Robert G. Wilson v, Aug 17 2011 STATUS approved

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Last modified May 12 04:27 EDT 2021. Contains 343810 sequences. (Running on oeis4.)