A068194 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.

1, 2, 3, 4, 7, 16, 31, 127, 256, 8191, 65536, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Author

Jon Perry, Feb 19 2002

Keywords

• nonn

Comments

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.

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

Consists of 1 and numbers m such that A001227(m) + A001227(m+1) = 3. - Juri-Stepan Gerasimov, Oct 06 2023

Examples

			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.
		

Links

• Jon Perry, Erdos-Moser.

Crossrefs

Cf. A000668, A001227, A019434, A068195.

A134459 is an essentially identical sequence.

Programs

Magma

[1] cat [m: m in [2..10000] | #Divisors(m)/Valuation(2*m, 2)+
#Divisors(m+1)/Valuation(2*(m+1),2) eq 3]; // Juri-Stepan Gerasimov, Oct 06 2023

Extensions

Edited by Dean Hickerson, Feb 22 2002