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A159273
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Smallest integer m, in absolute value, such that |m(prime(n)+m)| has as prime factors exactly all primes < sqrt(prime(n)); zero if no such m exists.
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1
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-1, 0, -1, 1, 1, -1, 1, -1, 1, 1, -1, 3, 4, 2, -2, 7, -14, 9, 3, -15, -3, 5, 7, -5, -7, 4, 2, -2, -4, 7, -22, -21, 28, 15, -44, 14, 63, 35, -35, 520, 910, 105, 264, 990, 1155, 165, 504, 1625, 2860, 156, 495, 91, 539, 750, 198, 507, 550, 275, 143, 720, 770, 2002, 2618
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OFFSET
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1,12
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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).
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PROG
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(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))}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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