%I #13 Aug 29 2020 02:47:25

%S 1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,

%T 1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,2,

%U 1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1,1,1,1,1,1,1

%N Square array read by falling antidiagonals, where A(n,k) = floor(A337472(n, k)/A337470(n, k)); Abundancy index of A337470(n, k) floored down.

%C Array is read by descending antidiagonals with n >= 0 and k >= 1 ranging as: (0, 1), (0, 2), (1, 1), (0, 3), (1, 2), (2, 1), (0, 4), (1, 3), (2, 2), (3, 1), ...

%H Antti Karttunen, <a href="/A337473/b337473.txt">Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F A(n,k) = floor(A337472(n, k)/A337470(n, k)).

%e The top left corner of the array begins as:

%e n/k | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

%e ----|----------------------------------------------------------------------

%e 0 | 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 1, 3, 2, 3, 3, 3, 3, 3,

%e 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2,

%e 2 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,

%e 3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

%e 4 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

%e 5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

%e etc.

%e A337474 gives the distance in each column to the first 1 in that column, being 0 for columns 1, 2, 4, 8, 16, ..., where 1 is already in the top row.

%o (PARI)

%o up_to = 105858-1; \\ Or 105-1.

%o A337473sq(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(floor(s))); prevpid = pid; e += f[i,2]); floor(s));

%o A337473list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(b=1, a, i++; if(i > #v, return(v)); v[i] = A337473sq(b-1, (a-(b-1))))); (v); };

%o v337473 = A337473list(up_to);

%o A337473(n) = v337473[1+n];

%Y Cf. A003961, A108951, A337470, A337472, A337474, A337476.

%K nonn,tabl

%O 0,4

%A _Antti Karttunen_, Aug 28 2020