Claims

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1990_01_01_borwein_loring: Borwein and Loring's 1990 identity writes n over two to the n as a sum of m consecutive terms k over two to the k for n = 2^(m+1) - m - 2, so infinitely many n have the property; refereed, it answers the first question yes.

1990_01_01_borwein_loring_conditional: Borwein and Loring's Corollary 1 shows that if the iteration a to 2(a mod n) always reaches zero (their Conjecture 1), then every n has the property of the second question; refereed and conditional, so it settles no standing.

2020_08_04_tengely_ulas_zygadlo: Tengely, Ulas and Zygadło verify that for every n at most ten thousand, n over two to the n is a sum of at least two distinct terms k over two to the k; refereed, it covers the second question up to ten thousand.

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