By Dan Slott
The Parkers vs. Regent.
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Additional resources for Amazing Spider-Man - Renew Your Vows 005 (2015)
3: It sufﬁces to show that both σ (a) = σ (b) and that σ (b) = a + b. Now σ (a) = σ (2 n )σ (p)σ (q) and σ (b) = σ (2 n )σ (r). Hence it sufﬁces to show that σ (p)σ (q) = σ (r)and σ (2 n )σ (r) = 2 n (pq + r). Since p, q, and r are prime, σ (p)σ (q) = (p + 1 )(q + 1 ) = 9 · 2 2n−1 = r + 1 = σ (r), establishing the ﬁrst equation. To establish the second equation, note that σ (2 n )σ (r) = (2 n+1 − 1 )(r + 1 ) = 2 n (9 · 2 2n − 9 · 2 n−1 ) = 2 n (9 · 2 2n − 6 · 2 n−1 − 3 · 2 n−1 ) = 2 n (9 · 2 2n−1 + 9 · 2 2n−1 − 3 · 2 n − 3 · 2 n−1 ) = 2 n (pq + r).
In all three cases, Player 1 moves to a winning position from which Player 2 is forced to move away. Consider the following example: Suppose Amy leaves the position (9, 15) for Andrew—namely two piles with 9 and 15 counters, respectively. Andrew has three possibilities. He can remove any number of counters from the 9’s pile, he can remove any number from the 15’s pile, or he can remove an equal number from both piles. Suppose he removes four counters from the 15’s pile, leaving Amy with position (9, 11).
10. Let s(n) = σ (n) − n denote the sum of the proper divisors of n. Deﬁne sk (n) recursively by s1 (n) = s(n) and sk (n) = s(sk−1 (n)) for k > 1. Call an integer n sociable of order k if n = sk (n) for some k ≥ 1. (a) Verify that perfect numbers are sociable of order 1 and amicable pairs are sociable of order 2. (b) Verify that 12,496 is sociable of order 5 (P. Poulet). List four other numbers sociable of order 5. ) 11. Prove the following two identities that appear in al-Banna’s Raf al-Hijab: (a) 13 + 33 + .