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

1,2

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.

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

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. A003415, A246278.

Cf. A068719 (row 1).

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

Sequence in context: A007309 A352186 A333341 * A084890 A058662 A344026

Adjacent sequences: A344024 A344025 A344026 * A344028 A344029 A344030

Keyword

nonn,tabl,look

Author

Antti Karttunen, May 07 2021

Status

approved