This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A316945 A triple of positive integers (n,p,k) is admissible if there exist at least two different multisets of k positive integers, {x_1,x_2,...,x_k} and {y_1,y_2,...,y_k}, such that x_1+x_2+...+x_k=y_1+y_2+...+y_k = n and x_1x_2...x_k = y_1y_2...y_k = p. For each n, let A(n)={k:(n,p,k) is admissible for some p}, and let a(n) = |A(n)|. 2
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 4, 6, 7, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,13 COMMENTS John Conway proposed an interesting math puzzle in the 1960s, which is now generally known as the "Conway's wizard problem." Here is the problem. Last night I sat behind two wizards on a bus and overheard the following: Blue Wizard: I have a positive integer number of children, whose ages are positive integers. The sum of their ages is the number of this bus, while the product is my own age. Red Wizard: How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages? Blue Wizard: No, you could not. Red Wizard: Aha! At last, I know how old you are! Apparently the Red Wizard had been trying to determine the Blue Wizard's age for some time. Now, what was the number of the bus? This problem posed by Conway looks at different multisets that correspond to the same ordered triple, which motivated the study of this sequence. LINKS Jay Bennett, Riddle of the week #34: Two wizards ride a bus, Popular Mechanics. Hearst Communications, Inc., 4 Aug. 2017. 12 Jun. 2018 Accessed. John B. Kelly, Partitions with equal products, Proc. Amer. Math. Soc. 15 (1964), 987-990. Tanya Khovanova, Conway's Wizards, arXiv:1210.5460 [math.HO], 2012. FORMULA a(n) = 0 for 1 <= n <= 11, a(12) = 1, a(13) = 2, a(14) = 4, a(15) = 4, a(16) = 6, a(17) = 7, a(18) = 7, and a(n) = n - 10 for n >= 19. EXAMPLE For n = 3, the only partitions of 3 are {3}, {1,2}, and {1,1,1}. Hence, there is no admissible triple (3,p,k). For n = 4, the only partitions of 4 are {4}, {1,3}, {2,2},{1,1,2}, and {1,1,1,1}. Hence, there is no admissible triple (4,p,k). For n = 12, the only admissible triple (12,p,k) is when p = 48 and k = 4. This is achieved by the following multisets: {1,3,4,4} and {2,2,2,6}. Thus a(12) = 1. MATHEMATICA Table[Count[   Table[intpart = IntegerPartitions[sum, {n}];    DuplicateFreeQ[     Table[Product[intpart[[i]][[j]], {j, n}], {i,       Length[intpart]}]], {n, sum}], False], {sum, 50}] CROSSREFS Cf. A060277, A316946. Sequence in context: A153494 A205404 A322372 * A307370 A014246 A026413 Adjacent sequences:  A316942 A316943 A316944 * A316946 A316947 A316948 KEYWORD nonn,easy AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 15 22:19 EDT 2019. Contains 324145 sequences. (Running on oeis4.)