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A068194 Numbers n for which the only representation of n(n+1)/2 as a sum of 2 or more consecutive positive integers is 1+2+...+n. 3
1, 2, 3, 4, 7, 16, 31, 127, 256, 8191, 65536, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Consists of 1, Mersenne primes (A000668) and Fermat primes (A019434) minus 1. Proof: The sum of r consecutive integers starting with a is r(r+2a-1)/2, so n(n+1)/2 has an extra representation of the desired form iff n(n+1)=rs where 1<r, r+1<s and r and s have opposite parity. If n is even, let n=2^e*m with m odd and let p be a prime divisor of n+1. Then we may take r=2^e and s=m(n+1) unless m=1 and we may take r=(n+1)/p and s=np unless n+1 is prime. Thus an even number n is in the sequence iff n+1 is a Fermat prime. Similarly an odd number n is in the sequence iff n=1 or n is a Mersenne prime.

Indices of partial maxima of A082184. - Ralf Stephan, Sep 01 2004

LINKS

Table of n, a(n) for n=1..18.

Jon Perry, Erdos-Moser

EXAMPLE

n=6 gives 21, which has the 2 representations 1+2+...+6 and 10+11, so 6 is not in the sequence. n=4 gives 10, whose only representation is 1+2+3+4, so 4 is in the sequence.

CROSSREFS

Cf. A068195. A134459 is an essentially identical sequence.

Sequence in context: A091155 A254432 A027362 * A217289 A179985 A110705

Adjacent sequences:  A068191 A068192 A068193 * A068195 A068196 A068197

KEYWORD

nonn

AUTHOR

Jon Perry, Feb 19 2002

EXTENSIONS

Edited by Dean Hickerson, Feb 22 2002

STATUS

approved

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Last modified July 20 12:33 EDT 2019. Contains 325180 sequences. (Running on oeis4.)