cp's OEIS Frontend

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

This is a subset of A003418 such that lcm(1,2,...,n-1) <> lcm(1,2,...,n) <> lcm(1,2,...,n+1) for (n>=1).

lcm(1,2,...,n) will be unique if both n and n+1 can be expressed as different prime powers, i.e., n = p^a and n+1 = q^b where p,q are prime and a,b are integers.