A356155 The pi-based arithmetic derivative applied to prime shift array: Square array A(n,k) = A258851(A246278(n,k)), read by falling antidiagonals.

1, 4, 2, 7, 12, 3, 12, 19, 30, 4, 11, 54, 41, 56, 5, 20, 26, 225, 79, 110, 6, 15, 87, 58, 588, 131, 156, 7, 32, 37, 310, 94, 1815, 193, 238, 8, 33, 216, 69, 861, 162, 3042, 269, 304, 9, 32, 140, 1500, 117, 2156, 218, 6069, 355, 414, 10, 21, 120, 427, 5488, 183, 3835, 314, 8664, 491, 580, 11, 52, 44, 455, 1254, 26620, 255, 6834, 422, 14283, 629, 682, 12

Offset

1,2

Comments

Each column is strictly 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
  2k =  2    4    6      8   10     12   14      16     18     20   22       24
-----+--------------------------------------------------------------------------
n= 1 |  1,   4,   7,    12,  11,    20,  15,     32,    33,    32,  21,      52,
   2 |  2,  12,  19,    54,  26,    87,  37,    216,   140,   120,  44,     351,
   3 |  3,  30,  41,   225,  58,   310,  69,   1500,   427,   455,  86,    2075,
   4 |  4,  56,  79,   588,  94,   861, 117,   5488,  1254,  1022, 132,    8183,
   5 |  5, 110, 131,  1815, 162,  2156, 183,  26620,  2561,  2717, 214,   31581,
   6 |  6, 156, 193,  3042, 218,  3835, 255,  52728,  4828,  4316, 304,   67093,
   7 |  7, 238, 269,  6069, 314,  6834, 373, 137564,  7695,  8075, 404,  154615,
   8 |  8, 304, 355,  8664, 422, 10241, 457, 219488, 12098, 12426, 524,  261003,
   9 |  9, 414, 491, 14283, 532, 17296, 609, 438012, 20909, 18653, 668,  535877,
  10 | 10, 580, 629, 25230, 718, 27231, 787, 975560, 29388, 31552, 836, 1050409,

Prog

(PARI)
up_to = 78;
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));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A356155sq(row, col) = A258851(A246278sq(row, col));
A356155list(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] = A356155sq(col, (a-(col-1))))); (v); };
v356155 = A356155list(up_to);
A356155(n) = v356155[n];

Crossrefs

Cf. A000027 (column 1), A097240 (column 3), A246278, A258851.

Cf. also A344027.

Sequence in context: A257502 A201207 A151890 * A227352 A255140 A108167

Adjacent sequences: A356152 A356153 A356154 * A356156 A356157 A356158

Keyword

nonn,tabl

Author

Antti Karttunen, Jul 29 2022

Status

approved