Sequence of length n having equal sum and
product
Louis Marmet
2005 March 19
(PAGE UNDER
CONSTRUCTION)
The series A104173(n),
is listed in the "OnLine Encyclopedia of Integer Sequences", is
defined as:
A104173(n) is the smallest integer
equal to the sum and the product of the same n positive integers:
A104173(n) = i(1) + i(2) + ... + i(n) = i(1) * i(2) * ... * i(n).
The first few terms are A104173(n)={1, 4, 6, 8, 8, 12, 12, 12, 15, 16, 16, 16, 18,
20, 24, 24, 24, 24, 24, 28, 27, 32, 30, 48, 32, 32, 32, 36, 36, 36, 42, 40, 40, 48, 48, 48,
45, 48, 48, 48, 48, 48, 54, 60, 54, 56, 54, 60, 63, 60, 60, 60, 63,
64, 64, 64, 64, 64, 70, 72, 72, 72, 72, 72, 72, 84, 80, 80, 81, 80,
80, 80, 81, 84, 88, 96, 90, 96, 90, 108, 90, 96, 96, 96, 96, 96, 96,
96, 96, 100, 110, 112, 105, 108, 108, 108, 117, 108, 108, 108, 112,
112, 120, 120, 120, 120, 120, 120, 120, 120, 120, 152, 125, 228, 126, 128, 128, 128, 128,
128, 128, 140, ...}.
The factors are (more numbers here, 1MB file!):
A104173( 1) = 1
= 1+ 1×0
A104173( 2) = 4
= 2+ 2+ 1×0
A104173( 3) = 6
= 3+ 2+ 1×1
A104173( 4) = 8
= 4+ 2+ 1×2
A104173( 5) = 8
= 2+ 2+ 2+ 1×2
A104173( 6) = 12 =
6+ 2+ 1×4
A104173( 7) = 12 =
4+ 3+ 1×5
A104173( 8) = 12 =
3+ 2+ 2+ 1×5
A104173( 9) = 15 =
5+ 3+ 1×7
A104173( 10)= 16 =
4+ 4+ 1×8
A104173( 11)= 16 =
4+ 2+ 2+ 1×8
A104173( 12)= 16 =
2+ 2+ 2+ 2+ 1×8
A104173( 13)= 18 =
3+ 3+ 2+ 1×10
A104173( 14)= 20 =
5+ 2+ 2+ 1×11
A104173( 15)= 24 =
8+ 3+ 1×13
A104173( 16)= 24 =
6+ 4+ 1×14
A104173( 17)= 24 =
6+ 2+ 2+ 1×14
A104173( 18)= 24 =
4+ 3+ 2+ 1×15
A104173( 19)= 24 =
3+ 2+ 2+ 2+ 1×15
A104173( 20)= 28 =
7+ 2+ 2+ 1×17
A104173( 21)= 27 =
3+ 3+ 3+ 1×18
A104173( 22)= 32 =
8+ 4+ 1×20
A104173( 23)= 30 =
5+ 3+ 2+ 1×20
A104173( 24)= 48 = 24+
2+ 1×22
...
A104173(114)=228 = 114+ 2+
1×112
...
A104173(174)=348 = 174+ 2+
1×172
...
A104173(444)=888 = 444+ 2+
1×442
Of interest is the series when A104173(n)=2n. This is the
series A033179(n)
representing the "Numbers i such that there is just one sequence of
length i having equal sum and product."
The series is A033179(n) = {2, 3, 4, 6, 24, 114, 174,
444}. No more terms are given. A search up to
13587782064 has not found any additional term.
Attempt to find another term for A033179
(Or to prove that the series
is finite!)
I. BASIC DEFINITIONS
Excluding the trivial case with n=1, we wish to find n
positive integers i_{1}, i_{2}, i_{3},
..., i_{n} such that their sum is equal to their product:
i_{1}+i_{2}+i_{3}+...
+i_{n} = i_{1}×i_{2}×i_{3}×...
×i_{n} [n>1].

(1)

To avoid repetition of
equivalent solutions, we sort the i_{k} such that i_{k}≥i_{k+1}.
For
example, the solutions for n<8 are:
n=2: 4= 2+2= 2×2
n=3: 6= 3+2+1= 3×2×1
n=4: 8= 4+2+1+1= 4×2×1×1
n=5: 10= 5+2+1+1+1= 5×2×1×1×1
9=
3+3+1+1+1= 3×3×1×1×1
8=
2+2+2+1+1= 2×2×2×1×1
n=6: 12= 6+2+1+1+1+1=
6×2×1×1×1×1
n=7: 14= 7+2+1+1+1+1+1=
7×2×1×1×1×1×1
12=
4+3+1+1+1+1+1= 4×3×1×1×1×1×1
If there is a solution to Eq. (1) and i_{2},
i_{3}, ..., i_{n} are given, there is only one
possible value for i_{1}. To demonstrate this, we
write Eq. (1) differently:
g(n,p) =
a×f_{1}×f_{2}×... ×f_{r}= a+f_{1}+f_{2}+...
+f_{r}+1×(nr1) [a≥f_{k}≥f_{k+1}≥2,
n>r≥1],

(2)

where a and f_{i} are
positive integers and p is the product of the f_{i}:

r
p = Π f_{i} [f_{i}≥2, r≥1].
i=1

(4)

It is useful to define s as
the sum of the f_{i}:

r
s = Σ f_{i} [f_{i}≥2, r≥1],
i=1 
(3)

and h=sr. From these it
is easy to see that the following useful relations hold:
p ≥ s ≥
2r and p ≥ s > h ≥ r.

(5)

We thus obtain the following
list for n<8:
g(2,2)= 4= 2+2= 2×2
g(3,2)= 6= 3+2+1= 3×2
g(4,2)= 8= 4+2+1+1= 4×2
g(5,2)= 10= 5+2+1+1+1= 5×2
g(5,3)= 9= 3+3+1+1+1=
3×3
g(5,4)= 8= 2+2+2+1+1=
2×2×2
g(6,2)= 12= 6+2+1+1+1+1= 6×2
g(7,2)= 14= 7+2+1+1+1+1+1= 7×2
g(7,3)= 12= 4+3+1+1+1+1+1= 4×3
II. GENERAL RELATIONS
THEOREM 1:
For given values of i_{2}, i_{3}, ..., i_{n},
there is either no solution or only one possible value of i_{1}
that will satisfy Eq. (1).
It is easy to show that for a solution of Eq. (2) to exist,
we must have:

n+h1
a = ——————— [a is an integer, p>1],
p1 
(6)

since (n+h1)÷(p1)×p=
(n+h1)÷(p1)+s+(nr1). We therefore have from Eq.
(2):

n+h1
g(n,p) = ———————×p.
p1

(7)

It follows that for given
values of i_{2}, i_{3}, ..., i_{n}, the
values for f_{1}, f_{2}, ..., f_{r}, s, p
and r are defined and there is a unique solution if:
n ≡ 0 (mod 1) if
p=2, n ≡ ph (mod p1) if p>2,

(8)

no solution otherwise.
Q.E.D.
THEOREM 2:
For given values of n, s and p, there could be more than one
representation for g(n,p), but all with the same value of r.
For example:
g(558,72)= 576= 8+6+6+2+1×554= 8×6×6×2 and g(558,72)= 576=
8+8+3+3+1×554= 8×8×3×3.
The only way to satisfy Eq. (8) with another value r' would
be to have n+sr'1 ≡ 0 (mod p1) with r'=r+m×(p1) [m a positive
integer]. From Eq. (5) p≥2r' we obtain the condition
p≥2(r+m(p1)) which is never satisfied. There is therefore
no other value of r satisfying Eq. (8).
Q.E.D.
THEOREM 3: For given values of n, r and
p, there could be more than one representation for g(n,p), but all
with the same value of s.
The only way to satisfy Eq. (8) with another value s' would
be to have n+s'r1 ≡ 0 (mod p1) with s'=s+m×(p1) [m a positive
integer]. From Eq. (5) p≥s' we obtain the condition
p≥s+m(p1) which is never satisfied. There is therefore no
other value of s satisfying Eq. (8).
Q.E.D.
Note that if n, r and s are given, there could be more than
one values of p that would satisfy Eq. (8). For example:
g(53,8)= 64= 8+4+2+1×50 but g(53,9)= 63= 7+3+3+1×50.
THEOREM 4:
For a given n and p, the value h=sr is the same for any solution
of Eq. (2). The value of g(n,p) is completely determined by
n and p.
For example:
g(275,48)= 288= 6+4+4+3+1×271, h=8, and g(275,48)= 288=
6+6+2+2+2+1×270, h=8.
We can simply rewrite Eq. (8) as: h+n ≡ 1 (mod p1).
There is no other solution of the form h'=h+m×(p1) since from Eq.
(5), h'<p. From Eq. (7) follows that the value of g(n,p)
is completely determined by n and p.
Q.E.D.
THEOREM 5:
For a given n, there is always a solution which is of the form:
n+2+1+... =
n×2×1×... = 2n.
Using Eq. (8) with r=1, s=2 and p=2, n ≡ 0 (mod 1), that
is, any n gives a solution to Eq. (1).
Q.E.D.
III. USEFUL RELATIONS
THEOREM 6:
Given a value for n, the largest value g(n,p) can take is 2n.
Since a=(n+h1)÷(p1) and a≥f_{1} (Eqs. (2) and
(6)), we find:
We want to confirm
that:
g(n,p)= a×p=

n+h1
———————×p ≤ 2n.
p1 
(10)

We can simplify Eq. (10) to
get:
Eq. (11) should hold true for
all values of n satisfying Eq. (9). We have then:
(h1)×p ≤
(f_{1}×(p1)h+1)×(p2).

(12)

A few simplifications of Eq.
(12) gives:
f_{1}×(p2)
≥
2×(h1). 
(13)

Starting from:
f_{b}×(p_{b}2)
≥
2×(h_{b}1) [b<r]. 
(14)

where: p_{b}=f_{b}×f_{b+1}×...×f_{r},
h_{b}=s_{b}r_{b}, s_{b}=f_{b}+f_{b+1}+...+f_{r}
and r_{b}=rb+1, we obtain:
f_{b}×(f_{b}p_{b+1}4)+2
≥
2×(h_{b+1}1).
Since f_{b}≥f_{b+1}≥2
and
p_{b+1}>1, it is easy to see that f_{b}×(f_{b}p_{b+1}4)+2 ≥
f_{b+1}×(p_{b+1}2). The relation:
f_{b}×(f_{b}p_{b+1}4)+2
≥
f_{b+1}×(p_{b+1}2) ≥ 2×(h_{b+1}1)
[b<r, p_{b+1}>1]
is true if:
f_{b+1}×(p_{b+1}2)
≥
2×(h_{b+1}1). 
(15)

Eq. (15) is Eq. (14) with the
first factor f_{b} removed.
It is easy to show that Eq. (14) is true for b=r since s_{r}=p_{r}=f_{r}
and h_{r}=f_{r}1:
f_{r}×(f_{r}2)
≥
2×(f_{r}2). 
(16)

This is true since f_{r}≥2.
We
now recursively show that if Eq. (16) is true, Eq. (14) is true
for b=r1, b=r2, etc. Eq. (14) is therefore true for b=1
which corresponds to Eq. (13). Thus Eq. (11) is true which
leads to:
Q.E.D.
IV. SOLUTIONS TO
g(n,p)<2n
Details about each step of the calculation are given at
this page: long.html.
... to be continued ...
Updated last: Nov. 10^{th},
2018
Visitors since March 28^{th}, 2005: