Hello Allan,
this is interesting ... thank you — and I agree! Have I got all that right? ... 100%, yes! more seqs ... I don’t want to bother Jean-Marc anymore, so if you want to compute and add them, please do (we will submit 2 variants today: — in the first one we « compact » the infinite sum of the primes with natural numbers (that are themselves the unique sum of consecutive primes, start of the seq is 5 (=2+3), 12 (=5+7), etc. — in the second one we compact the same sum _with prime numbers_ only (start is 5(=2+3), 23(=5+7+11) Best, É. Catapulté de mon aPhone
Le 15 août 2020 à 02:41, Allan Wechsler <acwacw@gmail.com> a écrit :
This is interesting. May I ask for a clarification?
I assume that this sequence is produced with a greedy algorithm; that is, having disposed of all the natural numbers below k, we incorporate the smallest sum k + ... + (k+n) that is the only nontrivial trapezoidal representation of that sum. If we reach a point where there is no such sum, then the sequence ends.
We do not backtrack, looking for the lexicographically earliest infinite sequence of unique sums.
We conjecture that the greedy sequence does not end.
Have I got all that right?
None of the obvious related sequences are in OEIS yet: the lengths of the blocks 2,2,2,3,4,2,16,..., the starting numbers 1,3,5,7,10,14,16,..., the ending numbers 2,4,6,9,13,15,31,....
I think a crucial insight is that the allowable sums are numbers with exactly one nontrivial odd divisor, that is, numbers of the form p*2^n, p>1. This means that there are two more associated sequences; the sequence of p (3,7,11,3,23,29,47,...) and the sequence of n (0,0,0,3,1,0,3,1,0,...); neither of these are in OEIS.
On Fri, Aug 14, 2020 at 7:47 PM Éric Angelini <eric.angelini@skynet.be> wrote:
Hello Math-fun, could I recommend to our members who like erratic behaviors to have a look at http://oeis.org/A336897 Best, É. Catapulté de mon aPhone
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