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%I #25 Oct 29 2023 21:47:25
%S 1,2,3,4,7,16,31,127,256,8191,65536,131071,524287,2147483647,
%T 2305843009213693951,618970019642690137449562111,
%U 162259276829213363391578010288127,170141183460469231731687303715884105727
%N 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.
%C 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.
%C Indices of partial maxima of A082184. - _Ralf Stephan_, Sep 01 2004
%C Consists of 1 and numbers m such that A001227(m) + A001227(m+1) = 3. - _Juri-Stepan Gerasimov_, Oct 06 2023
%H Jon Perry, <a href="https://web.archive.org/web/20060515222746/http://www.users.globalnet.co.uk/~perry/maths/erdosmoser/erdosmoser.htm">Erdos-Moser</a>.
%e k=6 gives 21, which has the 2 representations 1+2+...+6 and 10+11, so 6 is not in the sequence.
%e k=4 gives 10, whose only representation is 1+2+3+4, so 4 is in the sequence.
%o (Magma) [1] cat [m: m in [2..10000] | #Divisors(m)/Valuation(2*m, 2)+
%o #Divisors(m+1)/Valuation(2*(m+1),2) eq 3]; // _Juri-Stepan Gerasimov_, Oct 06 2023
%Y Cf. A000668, A001227, A019434, A068195.
%Y A134459 is an essentially identical sequence.
%K nonn
%O 1,2
%A _Jon Perry_, Feb 19 2002
%E Edited by _Dean Hickerson_, Feb 22 2002
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