A341608 Square array A(n,k) = A341524(A246278(n,k)), read by falling antidiagonals; number of prime factors (with mult.) in the denominator of abundancy index as applied onto prime shift array A246278.

1, 2, 1, 0, 2, 1, 3, 1, 2, 1, 1, 3, 2, 2, 1, 1, 2, 3, 2, 2, 1, 1, 2, 2, 3, 2, 2, 1, 4, 1, 3, 1, 3, 2, 2, 1, 2, 4, 2, 3, 2, 3, 2, 2, 1, 2, 3, 4, 2, 3, 2, 3, 2, 2, 1, 1, 3, 3, 4, 2, 3, 2, 3, 2, 2, 1, 1, 2, 3, 2, 4, 2, 3, 2, 3, 2, 2, 1, 1, 2, 2, 2, 3, 4, 2, 3, 2, 3, 2, 2, 1, 0, 1, 4, 2, 3, 3, 4, 2, 3, 2, 3, 2, 2, 1

Offset: 1

Author

Antti Karttunen, Feb 16 2021

Keywords

• nonn
• tabl

Examples

			The top left corner of the array:
   n=  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21
  2n=  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42
-----+---------------------------------------------------------------
   1 | 1, 2, 0, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 0, 1, 5, 1, 4, 1, 2, 1,
   2 | 1, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 2, 1, 2, 2, 5, 2, 4, 1, 4, 2,
   3 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 1, 3, 3, 5, 2, 4, 1, 4, 2,
   4 | 1, 2, 2, 3,*1, 3, 2, 4,*2,*2, 2, 4, 2, 3,*2, 5, 2,*3, 2,*3, 3,
   5 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2,*2, 3, 5, 2, 4, 2, 4, 3,
   6 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2,*3, 2, 3, 3, 5, 2, 4, 2, 4, 3,
   7 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
   8 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3,*1, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
   9 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  10 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  11 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2,*3, 2, 3, 3, 5, 2, 4,*1, 4, 3,
  12 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  13 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  14 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  15 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2,*2, 3, 5, 2, 4, 2, 4, 3,
  16 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  17 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  18 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  19 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  20 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
  21 | 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3,
etc.
Positions where columns are not monotonic (i.e., with sudden drops) are marked with an asterisk (*). See the example section of A341606 for their further elaboration.
		

Links

• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A001222, A017666, A341606, A341607, A341628.

Sequence A341524 applied to prime shift array A246278.

Programs

PARI

up_to = 105;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A017666(n) = denominator(sigma(n)/n);
A341608sq(row,col) = bigomega(A017666(A246278sq(row,col)));
A341608list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341608sq(col,(a-(col-1))))); (v); };
v341608 = A341608list(up_to);
A341608(n) = v341608[n];

Formula

A(n,k) = A001222(A341606(n,k)) = A001222(A017666(A246278(n,k))).