A337473 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.

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, 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, 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

Offset: 0

Author

Antti Karttunen, Aug 28 2020

Keywords

• nonn
• tabl

Comments

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), ...

Examples

			The top left corner of the array begins as:
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
----|----------------------------------------------------------------------
  0 | 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 1, 3, 2, 3, 3, 3, 3, 3,
  1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2,
  2 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,
  3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  4 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
etc.
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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to primorial numbers
• Index entries for sequences related to sigma(n)

Crossrefs

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

Programs

PARI

up_to = 105858-1; \\ Or 105-1.
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));
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); };
v337473 = A337473list(up_to);
A337473(n) = v337473[1+n];

Formula

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