A344027 Arithmetic derivative applied to prime shift array: Square array A(n,k) = A003415(A246278(n,k)), read by falling antidiagonals.

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

Author

Antti Karttunen, May 07 2021

Keywords

• nonn
• tabl
• look

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.

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003415, A246278.

Cf. A068719 (row 1).

Cf. also A343221, A343222, A344026, A355927, A356155.

Programs

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];