A334246 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) = {(p,k):(n,p,k) is admissible for some k}; then a(n) = |A(n)|.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 7, 13, 20, 29, 44, 66, 90, 129, 174, 232, 306, 406, 520, 675, 851, 1068, 1329, 1640, 2001, 2460, 2989, 3615, 4342, 5202, 6204, 7381, 8697, 10256, 12042, 14069, 16435, 19090, 22141, 25607, 29534

Offset: 1

Author

Bryan Bischof, Apr 20 2020

Keywords

• nonn

Comments

J. H. Conway proposed an interesting math puzzle in the 1960s, which is now generally known as "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.

The number n for which a(n)=1 gives the solution to Conway's puzzle.

Examples

			For n=12: {(4, 48)}.
For n=13: {(3, 36), (5, 48)}.
For n=14: {(4, 36), (3, 40), (3, 72), (5, 96), (6, 48)}.
For n=15: {(5, 36), (5, 144), (4, 72), (4, 40), (6, 96), (7, 48), (4, 96)}.
		

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.

Crossrefs

If only counting unique k: A316945; if only counting unique p: A316946.

See A140437 for another sequence related to the wizards problem.

Programs

Python

from collections import Counter
from functools import reduce
def partitions(n, I=1):
    yield (n,)
    for i in range(I, n//2 + 1):
        for p in partitions(n-i, i):
            yield (i,) + p
def p(i): #ret partitions of i, sorted by part number and product of parts
    return sorted(
        [
            (
                len(p),
                reduce(
                    (lambda x, y: x * y), p)
            )
            for p in partitions(i)
        ]
    )
def a(p_list): #returns number of pairs appearing more than once
    return len([x for x,y in Counter(p_list).most_common() if y > 1])
print(a(p(i))) # Will print the value of a(i)