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A344027
Arithmetic derivative applied to prime shift array: Square array A(n,k) = A003415(A246278(n,k)), read by falling antidiagonals.
9
1, 4, 1, 5, 6, 1, 12, 8, 10, 1, 7, 27, 12, 14, 1, 16, 10, 75, 18, 22, 1, 9, 39, 16, 147, 24, 26, 1, 32, 14, 95, 20, 363, 30, 34, 1, 21, 108, 18, 203, 28, 507, 36, 38, 1, 24, 55, 500, 24, 407, 32, 867, 42, 46, 1, 13, 51, 119, 1372, 30, 611, 40, 1083, 52, 58, 1, 44, 16, 135, 275, 5324, 36, 935, 48, 1587, 60, 62, 1
OFFSET
1,2
COMMENTS
For each column k, A343221(2*k) gives the least n (row number) where A(n,k) < A246278(n,k).
Each column is monotonic.
EXAMPLE
The top left corner of the array:
k = 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2k = 2 4 6 8 10 12 14 16 18 20 22 24 26 28
------+--------------------------------------------------------------------------
n=1 | 1, 4, 5, 12, 7, 16, 9, 32, 21, 24, 13, 44, 15, 32,
2 | 1, 6, 8, 27, 10, 39, 14, 108, 55, 51, 16, 162, 20, 75,
3 | 1, 10, 12, 75, 16, 95, 18, 500, 119, 135, 22, 650, 24, 155,
4 | 1, 14, 18, 147, 20, 203, 24, 1372, 275, 231, 26, 1960, 30, 287,
5 | 1, 22, 24, 363, 28, 407, 30, 5324, 455, 495, 34, 6050, 40, 539,
6 | 1, 26, 30, 507, 32, 611, 36, 8788, 731, 663, 42, 10816, 44, 767,
7 | 1, 34, 36, 867, 40, 935, 46, 19652, 1007, 1071, 48, 21386, 54, 1275,
8 | 1, 38, 42, 1083, 48, 1235, 50, 27436, 1403, 1463, 56, 31768, 60, 1539,
9 | 1, 46, 52, 1587, 54, 1863, 60, 48668, 2175, 1955, 64, 58190, 66, 2231,
10 | 1, 58, 60, 2523, 66, 2639, 70, 97556, 2759, 2987, 72, 102602, 76, 3219,
11 | 1, 62, 68, 2883, 72, 3255, 74, 119164, 3663, 3503, 78, 136462, 84, 3627,
12 | 1, 74, 78, 4107, 80, 4403, 84, 202612, 4715, 4551, 90, 219040, 96, 4847,
etc.
PROG
(PARI)
up_to = 91;
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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));
A344027sq(row, col) = A003415(A246278sq(row, col));
A344027list(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] = A344027sq(col, (a-(col-1))))); (v); };
v344027 = A344027list(up_to);
A344027(n) = v344027[n];
CROSSREFS
Cf. A068719 (row 1).
Sequence in context: A007309 A352186 A333341 * A084890 A058662 A344026
KEYWORD
nonn,tabl,look
AUTHOR
Antti Karttunen, May 07 2021
STATUS
approved