A341608 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..105.

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

FORMULA

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

EXAMPLE

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.

PROG

(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];

CROSSREFS

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

Sequence A341524 applied to prime shift array A246278.

Sequence in context: A258033 A153248 A221179 * A153247 A071432 A194508

Adjacent sequences: A341605 A341606 A341607 * A341609 A341610 A341611

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Feb 16 2021

STATUS

approved