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A068194
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Numbers k for which the only representation of k*(k+1)/2 as a sum of 2 or more consecutive positive integers is 1+2+...+k.
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8
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1, 2, 3, 4, 7, 16, 31, 127, 256, 8191, 65536, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
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OFFSET
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1,2
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COMMENTS
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Consists of 1, Mersenne primes (A000668) and Fermat primes (A019434) minus 1. Proof: The sum of r consecutive integers starting with j is r*(r + 2*j - 1)/2, so k*(k+1)/2 has an extra representation of the desired form iff k*(k+1) = r*s where 1 < r, r+1 < s, and r and s have opposite parity. If k is even, let k = 2^e*m with m odd and let p be a prime divisor of k+1. Then we may take r = 2^e and s = m*(k+1) unless m=1 and we may take r = (k+1)/p and s = k*p unless k+1 is prime. Thus an even number k is in the sequence iff k+1 is a Fermat prime. Similarly an odd number k is in the sequence iff k=1 or k is a Mersenne prime.
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LINKS
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EXAMPLE
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k=6 gives 21, which has the 2 representations 1+2+...+6 and 10+11, so 6 is not in the sequence.
k=4 gives 10, whose only representation is 1+2+3+4, so 4 is in the sequence.
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PROG
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(Magma) [1] cat [m: m in [2..10000] | #Divisors(m)/Valuation(2*m, 2)+
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CROSSREFS
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A134459 is an essentially identical sequence.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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