1,12

For prime(2)=3 no such integer exists, since there are no primes less than sqrt(3), but |m(3-m)|>1 for all nonzero m.

For all other primes up to prime(78)=397, a(n) is quickly calculated using the given code, with a(64)=19800 and a(68)=1724463 being the two largest values.

Since m and prime(n)-m cannot have a common prime factor, their respective prime factors form a partition of the primes <= sqrt(prime(n)). See A159273 for further details.

Several users at mersenneforum.org, A well-known puzzle..., February 2009.

a(n) = A159273(A000040(n)).

a(1)=-1 since |-1*(2-1)|=1 has no prime factors, i.e. exactly the primes less than sqrt(2): There are none.

a(2)=0 since there is no (nonzero) integer m such that |m*(3+m)| has as prime factors exactly the primes less than sqrt(3), i.e. none.

a(3)=-1 since prime(3)=5=1+2^2, thus |-1*(5-1)|=2^2 has {2}={primes < sqrt(5)} as prime factors.

a(4)=1 since prime(4)=7=2^3-1, thus |1*(7+1)|=2^3 has {2}={primes < sqrt(7)} as prime factors.

a(77) = 2926 since prime(77) = 389 = 3315 - 2926 = 3*5*13*17 - 2*7*11*19, thus |2926*(389+2926)| = product of all primes < sqrt(389).

(PARI) A159273(n)={ local(P=vector(primepi(sqrtint(n=prime(n))), i, prime(i))~, M); P|return(-(n==1)); M=P[ #P]; for( m=1, n-1, factor(m*(m+n))[, 1]==P & return(m); factor(m*(n-m))[, 1]==P & return(-m)); for( m=1+n, 9e9, vecmax(factor(m)[, 1])>M & next; factor(m*(m+n))[, 1]==P & return(m); factor(m*(m-n))[, 1]==P & return(-m))}

Sequence in context: A081246 A096411 A143486 this_sequence A021749 A088916 A117966

Adjacent sequences: A159270 A159271 A159272 this_sequence A159274 A159275 A159276

sign

M. F. Hasler (MHasler(AT)univ-ag.fr), Apr 09 2009