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 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
LINKS
Jon Perry, Erdos-Moser.
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon Perry, Feb 19 2002
EXTENSIONS
Edited by Dean Hickerson, Feb 22 2002
STATUS
approved