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A304657
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Number of ways to represent n/100 years in Shinigami time.
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0
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26, 76, 76, 208, 176, 252, 176, 544, 208, 604, 176, 768, 176, 604, 604, 1376, 176, 768, 176, 1888, 604, 604, 176, 2208, 818, 604, 544, 1888, 176, 2316, 176, 3392, 604, 604, 1460, 2568, 176, 604, 604, 5536, 176, 2316, 176, 1888, 1888, 604, 176, 6080, 818, 3172
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OFFSET
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1,1
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COMMENTS
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"Lifespan" and "Shinigami time", as used in this sequence, are concepts from the Death Note manga and anime series by Tsugumi Ohba and Takeshi Obata. The lifespan is represented in the series as a set of numbers floating above a person's head when that person is looked at through the Shinigami Eyes. For example, the lifespan of Light Yagami, the main character in the series, is represented as 9 3 31 2 6 3 9 in chapter 5 of the manga and episode 3 of the anime. (In the anime, it is unclear whether the 2 and the 6 are supposed to be connected or not. If they are treated as 26, the resulting number of years would be too large, see below). However, the exact meaning of these numbers is never revealed. Tsugumi Ohba, the author of the manga, supposedly said in an interview he created "a complicated math equation for Light's Life span when the numbers first appeared above his head", but later forgot what it was.
The following is my own theory regarding the meaning of the lifespan numbers of Light Yagami. I started by making an assumption: Light was supposed to live for about 70 to 90 years, neither extremely short, nor extremely long for a Japanese around the year 2000 (cf. WHO, p. 167). When one treats the lifespan as displayed in the manga as a sequence of numbers, computes their product and divides the number by 3600, one obtains 9 * 3 * 31 * 2 * 6 * 3 * 9 = 271188 and 271188/3600 = 75.33, which could be interpreted as meaning that Light was supposed to live for 75 years and 4 months. That number lies within the range 70-90 and also is too nice a number to be a coincidence in my opinion. This suggests that Ohba probably came up with the lifespan by calculating 75.33 * 3600 = 271188, factoring that number into 2 * 2 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 31, multiplying some of the primes to obtain 2 * 6 * 9 * 9 * 3 * 3 * 31 and then rearranging those factors.
With the above assumptions it is clear that, due to the commutative property of multiplication, a given lifespan in years can have more than one representation in Shinigami time.
Also, while it is unclear whether the lifespan numbers represent the original or remaining lifespan of an individual, rule no. 42 could be interpreted as meaning that the lifespan of Light that Ryuk sees is the original lifespan that probably never changes as Light ages (cf. Death Note Wiki, Manga Chapter Rules - Rule #42).
It seems the terms of the sequence are also terms of A034776. Why?
0.01 years, equal to roughly 3.5 days or 5256 minutes, is the shortest timespan that can be represented in Shinigami time. - Felix Fröhlich, Apr 04 2021
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REFERENCES
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Tsugumi Ohba and Takeshi Obata, Death Note 13: How to Read, Paw Prints, 2008.
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LINKS
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EXAMPLE
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For n = 1, we have 0.01*3600 = 36 and 36 = 2 * 2 * 3 * 3.
Thus, the possible representations are as follows:
2 2 3 3; 2 3 2 3; 2 3 3 2; 3 2 2 3; 3 2 3 2; 3 3 2 2
2 2 9; 2 3 6; 2 6 3; 2 9 2; 3 2 6; 3 3 4; 3 4 3; 3 6 2; 4 3 3; 6 2 3; 6 3 2; 9 2 2
2 18; 3 12; 4 9; 6 6; 9 4; 12 3; 18 2
36
There are 6 representations of length 4, 12 representations of length 3, 7 representations of length 2 and 1 representation of length 1. 6 + 12 + 7 + 1 = 26, so a(1) = 26.
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PROG
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(PARI) step_a(n) = n/100
step_b(n) = floor(n*3600)
step_c(n, o=[1])=if(n>1, concat(apply(t->vector(t[2], i, t[1]), Vec(factor(n)~))), o) \\ after M. F. Hasler in A027746
find_index_a(vec) = my(r=#vec-1); while(1, if(vec[r] < vec[r+1], return(r)); r--; if(r==0, return(-1)))
find_index_b(r, vec) = my(s=#vec); while(1, if(vec[r] < vec[s], return(s)); s--; if(s==r, return(-1)))
switch_elements(vec, firstpos, secondpos) = my(g); g=vec[secondpos]; vec[secondpos]=vec[firstpos]; vec[firstpos] = g; vec \\ from David A. Corneth
reverse_order(vec, r) = my(v=[], w=[]); for(x=1, r, v=concat(v, vec[x])); for(y=r+1, #vec, w=concat(w, vec[y])); w=Vecrev(w); concat(v, w)
next_permutation(vec) = my(r=find_index_a(vec)); if(r==-1, return(0), my(s=find_index_b(r, vec)); vec=switch_elements(vec, r, s); vec=reverse_order(vec, r)); vec
multiply_neighboring_elements(vec, pos) = my(x=vec[pos]*vec[pos+1], w=[]); for(k=1, pos-1, w=concat(w, vec[k])); w=concat(w, [x]); for(r=pos+2, #vec, w=concat(w, vec[r])); w
all_permutations(vec) = my(v=[vecsort(vec)], w=vecsort(vec)); while(1, w=next_permutation(w); if(w==0, return(v), v=concat(v, [0]); v[#v]=w))
all_sets_from_multiplying_neighboring_elements(vec) = my(v=[]); for(k=1, #vec-1, v=concat(v, [0]); v[#v]=multiply_neighboring_elements(vec, k)); v
count_eligible_sets(vec) = my(v=all_permutations(vec), w=[], i=#v); while(1, for(x=1, #v, my(asfmne=all_sets_from_multiplying_neighboring_elements(v[x])); if(#asfmne==0, return(i)); for(y=1, #asfmne, w=concat(w, [0]); w[#w]=asfmne[y]); w=vecsort(w, , 8)); i=i+#w; v=w; w=[])
a(n) = my(f=step_c(step_b(step_a(n)))); count_eligible_sets(f)
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CROSSREFS
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KEYWORD
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dumb,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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