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A304657 Number of ways to represent n/100 years in Shinigami time.

%I A304657 #37 May 15 2021 12:36:20
%S A304657 26,76,76,208,176,252,176,544,208,604,176,768,176,604,604,1376,176,
%T A304657 768,176,1888,604,604,176,2208,818,604,544,1888,176,2316,176,3392,604,
%U A304657 604,1460,2568,176,604,604,5536,176,2316,176,1888,1888,604,176,6080,818,3172
%N A304657 Number of ways to represent n/100 years in Shinigami time.
%C A304657 "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.
%C A304657 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.
%C A304657 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.
%C A304657 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).
%C A304657 It seems the terms of the sequence are also terms of A034776. Why?
%C A304657 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
%D A304657 Tsugumi Ohba and Takeshi Obata, Death Note 13: How to Read, Paw Prints, 2008.
%H A304657 Death Note Wiki, <a href="http://deathnote.wikia.com/wiki/Lifespan">Lifespan</a>
%H A304657 Death Note Wiki, <a href="http://deathnote.wikia.com/wiki/Rules_of_the_Death_Note/Manga_Chapter_Rules#Volume_8">Rules of the Death Note/Manga Chapter Rules - Volume 8</a>
%H A304657 WHO, <a href="http://www.who.int/whr/2003/en/whr03_en.pdf">The World Health Report 2003</a>
%H A304657 Wikipedia, <a href="https://en.wikipedia.org/wiki/Death_Note">Death Note</a>
%e A304657 For n = 1, we have 0.01*3600 = 36 and 36 = 2 * 2 * 3 * 3.
%e A304657 Thus, the possible representations are as follows:
%e A304657 2 2 3 3; 2 3 2 3; 2 3 3 2; 3 2 2 3; 3 2 3 2; 3 3 2 2
%e A304657 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
%e A304657 2 18; 3 12; 4 9; 6 6; 9 4; 12 3; 18 2
%e A304657 36
%e A304657 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.
%o A304657 (PARI) step_a(n) = n/100
%o A304657 step_b(n) = floor(n*3600)
%o A304657 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
%o A304657 find_index_a(vec) = my(r=#vec-1); while(1, if(vec[r] < vec[r+1], return(r)); r--; if(r==0, return(-1)))
%o A304657 find_index_b(r, vec) = my(s=#vec); while(1, if(vec[r] < vec[s], return(s)); s--; if(s==r, return(-1)))
%o A304657 switch_elements(vec, firstpos, secondpos) = my(g); g=vec[secondpos]; vec[secondpos]=vec[firstpos]; vec[firstpos] = g; vec \\ from _David A. Corneth_
%o A304657 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)
%o A304657 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
%o A304657 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
%o A304657 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))
%o A304657 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
%o A304657 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=[])
%o A304657 a(n) = my(f=step_c(step_b(step_a(n)))); count_eligible_sets(f)
%Y A304657 Cf. A001055, A034776, A047726.
%K A304657 dumb,nonn
%O A304657 1,1
%A A304657 _Felix Fröhlich_, May 16 2018
%E A304657 Entry revised by _Felix Fröhlich_, Apr 03 2021