By Doron Zeilberger and Noam Zeilberger
Not all people, or combinatorial objects, are created equal. Hence it makes sense to count them by `importance', or `significance', rather than treat them all equally.
Added Oct. 31, 2018: Will Sawin, Columbia University, has solved (very quickly!) Question 1. A donation to the OEIS Foundation, in his honor has been made.
Added 10:30am, Nov. 2, 2018: Fernando Chamizo was the second to answer question 1, and his proof is even shorter.
Added 2:30pm, Nov. 2, 2018: Gjergji Zaimi kindly pointed out that Question 1 was posed and answered completely by D.H. Lehmer in Theorem 3 of his article On reciprocally weighted partitions, Acta Arithmetica XXI(1972), p. 379-388.
Added Nov. 5, 2018: Christopher Ryba (assisted by conversations with Pavel Etingof and Andrew Ahn) has answered Question 2. A donation of 100,ドル to the OEIS, in his honor has been made.
Added Nov. 6, 2018: Laurent Habsieger indpendently answered question 1.
Added Nov. 13, 2018: Apparently Question 1 is very natural. It was rediscovered (with full proof) in 1993 by Arnold Knopfmacher and J.N. Ridely in this article that appeared in SIAM J. Discrete Math, v. 6 (1993), 388-399 (that contains many other interesting results!). Who knows, perhaps it was done even before Lehmer's 1972 paper?
Articles of Doron Zeilberger