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The smallest k such that the product k*(k+1) is divisible by the first n primes and no others.
2

%I #35 Jul 10 2024 05:00:40

%S 1,2,5,14,384,1715,714,633555

%N The smallest k such that the product k*(k+1) is divisible by the first n primes and no others.

%C a(9)-a(21) do not exist. It seems unlikely that a(n) exists for larger n. [_Charles R Greathouse IV_, Aug 18 2011]

%C If a term beyond a(8) exists, it is larger than 2.29*10^25. - _Giovanni Resta_, Nov 30 2019

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_358.htm">Puzzle 358. Ruth-Aaron pairs revisited</a>, The Prime Puzzles & Problems Connection.

%e n smallest k k*(k+1) prime factorization

%e 1 1 2

%e 2 2 2*3

%e 3 5 2*3*5

%e 4 14 2*3*5*7

%e 5 384 2^7*3*5*7*11

%e 6 1715 2^2*3*7^3*11*13

%e 7 714 2*3*5*7*11*13*17

%e 8 633555 2^2*3^3*5*7*11^3*13*17*19^2

%t 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]

%t (* 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}]

%o (PARI) a(n)={

%o my(v=[Mod(0,1)],u,P=1,t,g,k);

%o forprime(p=2,prime(n),

%o P*=p;

%o u=List();

%o for(i=1,#v,

%o listput(u,chinese(v[i],Mod(-1,p)));

%o listput(u,chinese(v[i],Mod(0,p)))

%o );

%o v=0;v=Vec(u)

%o );

%o v=vecsort(lift(v));

%o while(1,

%o for(i=1,#v,

%o t=(v[i]+k)*(v[i]+k+1)/P;

%o if(!t,next);

%o while((g=gcd(P,t))>1, t/=g);

%o if (t==1, return(v[i]+k))

%o );

%o k += P

%o )

%o }; \\ _Charles R Greathouse IV_, Aug 18 2011

%o (Haskell)

%o a193314 n = head [k | k <- [1..], let kk' = a002378 k,

%o mod kk' (a002110 n) == 0, a006530 kk' == a000040 n]

%o -- _Reinhard Zumkeller_, Jun 14 2015

%Y Cf. A006145, A039945.

%Y Cf. A002110, A002378, A006530, A118478.

%K nonn

%O 1,2

%A _Robert G. Wilson v_, Aug 17 2011