A246278 -id:A246278 - cp's OEIS frontend

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

Antti Karttunen, May 07 2021

For each column k, A343221(2*k) gives the least n (row number) where A(n,k) < A246278(n,k).

Each column is monotonic.

			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.

Cf. A003415, A246278.
Cf. A068719 (row 1).
Cf. also A343221, A343222, A344026, A355927, A356155.

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];

A355925 Square array A(n, k) = A009194(A246278(n, k)), read by falling antidiagonals.

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 15, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 22 2022

			The top left corner of the array:
   k=  1  2  3  4  5  6  7  8  9 10  11  12 13  14 15 16 17 18  19  20 21
  2k=  2  4  6  8 10 12 14 16 18 20  22  24 26  28 30 32 34 36  38  40 42
-----+-----------------------------------------------------------------------
   1 | 1, 1, 6, 1, 2, 4, 2, 1, 3, 2,  2, 12, 2, 28, 6, 1, 2, 1,  2, 10, 6,
   2 | 1, 1, 3, 1, 1, 3, 3, 1, 1, 1,  1, 15, 3,  3, 3, 1, 1, 1,  3,  1, 3,
   3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 5,  1, 1, 1, 1, 1,  5,  1, 7,
   4 | 1, 1, 1, 1, 7, 1, 1, 1, 7, 7,  1,  1, 1,  1, 7, 1, 1, 7,  1,  7, 1,
   5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1, 19, 1, 1, 1, 1,  1,  1, 1,
   6 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 17, 1,  1, 1, 1, 1, 1,  1,  1, 1,
   7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
   8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
   9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 37, 1,  1, 1, 1, 1, 1, 31,  1, 1,
  12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1, 61, 1, 1, 1, 1,  1,  1, 1,
  16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  17 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  18 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  19 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  20 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
  21 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,

Cf. A000203, A009194, A246278.

Cf. also A341605, A341606, A341607, A341608, A341626, A341627, A355924, A355927 for related arrays of similar construction.

up_to = 105;
A009194(n) = gcd(n, sigma(n));
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));
A355925sq(row,col) = A009194(A246278sq(row,col));
A355925list(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] = A355925sq(col,(a-(col-1))))); (v); };
v355925 = A355925list(up_to);
A355925(n) = v355925[n];

A(n, k) = A009194(A246278(n, k)).
A(n, k) = gcd(A246278(n,k), A355927(n, k)).
A(n, k) = A355927(n, k) / A341605(n, k).
A(n, k) = A246278(n, k) / A341606(n, k).

A341607 Square array A(n,k) = A006530(A017666(A246278(n,k))), read by falling antidiagonals.

Original entry on oeis.org

2, 2, 3, 1, 3, 5, 2, 5, 5, 7, 5, 3, 7, 7, 11, 3, 7, 5, 11, 11, 13, 7, 5, 11, 7, 13, 13, 17, 2, 11, 7, 13, 11, 17, 17, 19, 3, 3, 13, 11, 17, 13, 19, 19, 23, 5, 5, 5, 17, 13, 19, 17, 23, 23, 29, 11, 7, 7, 7, 19, 17, 23, 19, 29, 29, 31, 2, 13, 11, 11, 11, 23, 19, 29, 23, 31, 31, 37, 13, 3, 17, 13, 13, 13, 29, 23, 31, 29, 37, 37, 41

Offset: 1

Antti Karttunen, Feb 16 2021

			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
  2n=   2   4   6   8  10  12  14  16  18  20  22  24   26  28  30  32   34
-----+----------------------------------------------------------------------
   1 |  2,  2,  1,  2,  5,  3,  7,  2,  3,  5, 11,  2,  13,  1,  5,  2,  17,
   2 |  3,  3,  5,  3,  7,  5, 11,  3,  5,  7, 13,  3,  17, 11,  7,  3,  19,
   3 |  5,  5,  7,  5, 11,  7, 13,  5,  7, 11, 17,  7,  19, 13, 11,  5,  23,
   4 |  7,  7, 11,  7, 13, 11, 17,  7, 11, 13, 19, 11,  23, 17, 13,  7,  29,
   5 | 11, 11, 13, 11, 17, 13, 19, 11, 13, 17, 23, 13,  29,*11, 17, 11,  31,
   6 | 13, 13, 17, 13, 19, 17, 23, 13, 17, 19, 29,*13,  31, 23, 19, 13,  37,
   7 | 17, 17, 19, 17, 23, 19, 29, 17, 19, 23, 31, 19,  37, 29, 23, 17,  41,
   8 | 19, 19, 23, 19, 29, 23, 31, 19, 23, 29, 37, 23,  41, 31, 29, 19,  43,
   9 | 23, 23, 29, 23, 31, 29, 37, 23, 29, 31, 41, 29,  43, 37, 31, 23,  47,
  10 | 29, 29, 31, 29, 37, 31, 41, 29, 31, 37, 43, 31,  47, 41, 37, 29,  53,
  11 | 31, 31, 37, 31, 41, 37, 43, 31, 37, 41, 47,*31,  53, 43, 41, 31,  59,
  12 | 37, 37, 41, 37, 43, 41, 47, 37, 41, 43, 53, 41,  59, 47, 43, 37,  61,
  13 | 41, 41, 43, 41, 47, 43, 53, 41, 43, 47, 59, 43,  61, 53, 47, 41,  67,
  14 | 43, 43, 47, 43, 53, 47, 59, 43, 47, 53, 61, 47,  67, 59, 53, 43,  71,
  15 | 47, 47, 53, 47, 59, 53, 61, 47, 53, 59, 67, 53,  71, 47, 59, 47,  73,
  16 | 53, 53, 59, 53, 61, 59, 67, 53, 59, 61, 71, 59,  73, 67, 61, 53,  79,
  17 | 59, 59, 61, 59, 67, 61, 71, 59, 61, 67, 73, 61,  79, 71, 67, 59,  83,
  18 | 61, 61, 67, 61, 71, 67, 73, 61, 67, 71, 79, 67,  83, 73, 71, 61,  89,
  19 | 67, 67, 71, 67, 73, 71, 79, 67, 71, 73, 83, 71,  89, 79, 73, 67,  97,
  20 | 71, 71, 73, 71, 79, 73, 83, 71, 73, 79, 89, 73,  97, 83, 79, 71, 101,
  21 | 73, 73, 79, 73, 83, 79, 89, 73, 79, 83, 97, 79, 101, 89, 83, 73, 103,
etc.
Positions where columns are not strictly monotonic are marked with an asterisk (*). See the example section of A341606 for further elaboration.

Cf. A006530, A017666, A246278, A341606, A341608, A341628.

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));
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A017666(n) = denominator(sigma(n)/n);
A341607sq(row,col) = A006530(A017666(A246278sq(row,col)));
A341607list(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] = A341607sq(col,(a-(col-1))))); (v); };
v341607 = A341607list(up_to);
A341607(n) = v341607[n];

A(n,k) = A006530(A341606(n, k)) = A006530(A017666(A246278(n,k))).

A341626 Square array A(n,k) = A341526(A246278(n,k)), read by falling antidiagonals; Numerators of the columnwise first quotients of A341605/A341606.

Original entry on oeis.org

8, 52, 9, 4, 279, 20, 64, 6, 1425, 21, 160, 1053, 10, 343, 77, 26, 189, 12500, 49, 22143, 117, 28, 372, 110, 62769, 33, 51883, 170, 1936, 231, 4275, 351, 791945, 130, 110109, 114, 248, 5751, 780, 2401, 6545, 573417, 68, 199633, 115, 1040, 2565, 1750625, 595, 199287, 13338, 1778506, 57, 460759, 464

Offset: 1

Antti Karttunen, Feb 16 2021

See comments in A341605.

			The top left corner of the array:
   n =  1       2    3        4      5        6      7             8        9
  2n =  2       4    6        8     10       12     14            16       18
----+--------------------------------------------------------------------------
  1 |   8,     52,   4,      64,   160,      26,    28,         1936,     248,
  2 |   9,    279,   6,    1053,   189,     372,   231,         5751,    2565,
  3 |  20,   1425,  10,   12500,   110,    4275,   780,      1750625,     980,
  4 |  21,    343,  49,   62769,   351,    2401,   595,     38668105,    6039,
  5 |  77,  22143,  33,  791945,  6545,  199287,  1463,    453007181,  307307,
  6 | 117,  51883, 130,  573417, 13338,  518830, 13455,   2534531701,  757809,
  7 | 170, 110109,  68, 1778506,  9775,  660654, 15776,  11489232281, 1786190,
  8 | 114, 199633,  57, 2181162, 17632,  998165, 33573,  38126842081, 2283762,
  9 | 115, 460759,  92, 5122307, 67735, 7372144, 89355, 204995005981, 3311655,
etc.

Cf. A246278, A341605, A341606, A341526.

Cf. A341627 (denominators).

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));
A341626sq(row,col) = A341526(A246278sq(row,col));
A341626list(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] = A341626sq(col,(a-(col-1))))); (v); };
v341626 = A341626list(up_to);
A341626(n) = v341626[n];

A(n,k) = A341526(A246278(n,k)).
If we set r(row,col) = A341605(row,col)/A341606(row,col) and d(row,col) = A(row,col)/A341627(row,col), then d(row,col) = r(row+1,col)/r(row,col).
For all n, k, A(n,k) < A341627(n, k).

A378979 Square array A(n, k) = 2*A246278(n, k) - sigma(A246278(n, k)), read by falling antidiagonals. Deficiency applied to the prime shift array.

Original entry on oeis.org

1, 1, 2, 0, 5, 4, 1, 6, 19, 6, 2, 14, 22, 41, 10, -4, 10, 94, 58, 109, 12, 4, 12, 38, 286, 118, 155, 16, 1, 18, 102, 70, 1198, 190, 271, 18, -3, 41, 46, 394, 158, 2014, 286, 341, 22, -2, 26, 469, 94, 1284, 214, 4606, 394, 505, 28, 8, 22, 148, 2001, 178, 2452, 350, 6478, 614, 811, 30, -12, 22, 178, 630, 13177, 262, 4842, 502, 11614, 838, 929, 36

Offset: 1

Antti Karttunen, Dec 13 2024

Each column is strictly increasing.

For all k >= 1, A(1+A378985(k), k) > 0, and it is the topmost positive number of the column k.

			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
----+-------------------------------------------------------------------
  1 |  1,   1,   0,    1,   2,   -4,   4,     1,   -3,   -2,   8,   -12,
  2 |  2,   5,   6,   14,  10,   12,  18,    41,   26,   22,  22,    30,
  3 |  4,  19,  22,   94,  38,  102,  46,   469,  148,  178,  62,   502,
  4 |  6,  41,  58,  286,  70,  394,  94,  2001,  630,  476, 106,  2746,
  5 | 10, 109, 118, 1198, 158, 1284, 178, 13177, 1522, 1720, 218, 14110,
  6 | 12, 155, 190, 2014, 214, 2452, 262, 26181, 3216, 2762, 334, 31858,
  7 | 16, 271, 286, 4606, 350, 4842, 446, 78301, 5416, 5926, 478, 82294,

Cf. A033879, A246278, A336835, A355927, A372562, A372563.
Cf. A006093 (column 1), A306190 (column 2), A378978 (row 1), A378985 (row index of the topmost positive term in column n).
Cf. also arrays A341605, A341606 and A341607.
Cf. also A324055.

up_to = 78;
A033879(n) = (n+n-sigma(n));
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));
A378979sq(row,col) = A033879(A246278sq(row,col));
A378979list(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] = A378979sq(col,(a-(col-1))))); (v); };
v378979 = A378979list(up_to);
A378979(n) = v378979[n];

A(n, k) = A033879(A246278(n, k)) = 2*A246278(n, k) - A355927(n, k).

A(n, k) = A372563(n,k) - A372562(n, k).

A379010 Square array A(n, k) = phi(A246278(n, k)), read by falling antidiagonals; Euler totient function applied to the prime shift array.

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 4, 8, 20, 6, 4, 18, 24, 42, 10, 4, 12, 100, 60, 110, 12, 6, 24, 40, 294, 120, 156, 16, 8, 20, 120, 72, 1210, 192, 272, 18, 6, 54, 48, 420, 160, 2028, 288, 342, 22, 8, 40, 500, 96, 1320, 216, 4624, 396, 506, 28, 10, 36, 168, 2058, 180, 2496, 352, 6498, 616, 812, 30, 8, 24, 200, 660, 13310, 264, 4896, 504, 11638, 840, 930, 36

Offset: 1

Antti Karttunen, Dec 14 2024

Each column is strictly increasing.

			The top left corner of the array:
k=  |  1     2     3      4     5      6     7        8      9     10
2k= |  2     4     6      8    10     12    14       16     18     20
----+-------------------------------------------------------------------
1   |  1,    2,    2,     4,    4,     4,    6,       8,     6,     8,
2   |  2,    6,    8,    18,   12,    24,   20,      54,    40,    36,
3   |  4,   20,   24,   100,   40,   120,   48,     500,   168,   200,
4   |  6,   42,   60,   294,   72,   420,   96,    2058,   660,   504,
5   | 10,  110,  120,  1210,  160,  1320,  180,   13310,  1560,  1760,
6   | 12,  156,  192,  2028,  216,  2496,  264,   26364,  3264,  2808,
7   | 16,  272,  288,  4624,  352,  4896,  448,   78608,  5472,  5984,
8   | 18,  342,  396,  6498,  504,  7524,  540,  123462,  9108,  9576,
9   | 22,  506,  616, 11638,  660, 14168,  792,  267674, 17864, 15180,
10  | 28,  812,  840, 23548, 1008, 24360, 1120,  682892, 26040, 29232,
11  | 30,  930, 1080, 28830, 1200, 33480, 1260,  893730, 39960, 37200,
12  | 36, 1332, 1440, 49284, 1512, 53280, 1656, 1823508, 59040, 55944,

Cf. A000010, A246278, A379011.

Cf. A062570 (row 1), A006093 (column 1), A036689 (column 2), A083553 (column 3), A135177 (column 4).

up_to = 11325; \\ = binomial(150+1,2)
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));
A379010sq(row,col) = eulerphi(A246278sq(row,col));
A379010list(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] = A379010sq(col,(a-(col-1))))); (v); };
v379010 = A379010list(up_to);
A379010(n) = v379010[n];

A246279 Transpose of array A246278: A(row,1) = 2*row, and for col > 1, A(row,col) = A003961(A(row,col-1)).

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 15, 8, 11, 49, 35, 27, 10, 13, 121, 77, 125, 21, 12, 17, 169, 143, 343, 55, 45, 14, 19, 289, 221, 1331, 91, 175, 33, 16, 23, 361, 323, 2197, 187, 539, 65, 81, 18, 29, 529, 437, 4913, 247, 1573, 119, 625, 75, 20, 31, 841, 667, 6859, 391, 2873, 209, 2401, 245, 63, 22

Offset: 1

Antti Karttunen, Aug 21 2014

Permutation of natural numbers larger than 1.

See comments in A246278 (the same array transposed).

			The top-left corner of the array:
   2,     3,     5,     7,    11,    13,    17,    19,    23, ...
   4,     9,    25,    49,   121,   169,   289,   361,   529, ...
   6,    15,    35,    77,   143,   221,   323,   437,   667, ...
   8,    27,   125,   343,  1331,  2197,  4913,  6859, 12167, ...
  10,    21,    55,    91,   187,   247,   391,   551,   713, ...
  12,    45,   175,   539,  1573,  2873,  5491,  8303, 15341, ......

Transpose of A246278.
One more than A246273.
Cf. A003961, A083140.

(define (A246279 n) (A246278bi (A004736 n) (A002260 n)))
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

If col = 1, A(row,col) = 2*row, otherwise A(row,col) = A003961(A(row,col-1)).

A253551 Square array: A(row,col) = 2^(row-1) * 1+(2*A156552(col)) = A156552(A246278(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 9, 14, 20, 24, 16, 11, 18, 28, 40, 48, 32, 17, 22, 36, 56, 80, 96, 64, 15, 34, 44, 72, 112, 160, 192, 128, 13, 30, 68, 88, 144, 224, 320, 384, 256, 19, 26, 60, 136, 176, 288, 448, 640, 768, 512, 33, 38, 52, 120, 272, 352, 576, 896, 1280, 1536, 1024, 23, 66, 76, 104, 240, 544, 704, 1152, 1792, 2560, 3072, 2048, 65, 46

Offset: 1

Antti Karttunen, Jan 03 2015

Shares with A135764 the property that A001511(n) = k for all terms n on row k and when going downwards in each column, terms grow by doubling.

			The top left corner of the array:
   1,  3,  5,   7,   9, 11,  17,  15,  13,  19,  33,  23,  65,  35,  21,
   2,  6, 10,  14,  18, 22,  34,  30,  26,  38,  66,  46, 130,  70,  42,
   4, 12, 20,  28,  36, 44,  68,  60,  52,  76, 132,  92, 260, 140,  84,
   8, 24, 40,  56,  72, 88, 136, 120, 104, 152, 264, 184, 520, 280, 168,
  16, 48, 80, 112, 144,176, 272, 240, 208, 304, 528, 368,1040, 560, 336,
...

Inverse: A253552.
Cf. A000079, A001511, A002260, A005408, A004736, A005940, A005941, A156552, A246278.
Differs from A135764 for the first time at n=22, where a(22) = 17, while A135764(22) = 13.

A(row,col) = A156552(A246278(row,col)).
A(row,col) = A135764(row, A005941(col)). [Is otherwise the same array as A135764, but the column positions have been permuted by A005941.]
A(row,col) = 2^(row-1) * ((2*A005941(col)) - 1) = 2^(row-1) * A005408(A156552(col)). [The above expands to this.]
As a composition of other permutations:
a(n) = A156552(A246278(n+1)). [When all three sequences are interpreted as one-dimensional sequences.]

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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 16 2021

			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.

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

Sequence A341524 applied to prime shift array A246278.

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];

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

A355924 Square array A(n,k) = A342671(A246278(n,k)), read by falling antidiagonals, where A342671(x) = gcd(sigma(x), A003961(x)).

Original entry on oeis.org

3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 17, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 37, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 21 2022

			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 | 3, 1, 3,  3, 3, 1, 3,  1, 3, 21, 3, 15,  3,  1, 3, 9, 3, 1, 3,  9, 3,
   2 | 1, 1, 1,  5, 1, 1, 1,  1, 1,  1, 1,  5,  1, 13, 1, 1, 5, 1, 1,  5, 1,
   3 | 1, 1, 1,  1, 1, 1, 7,  1, 1,  1, 1,  1,  1,  7, 1, 7, 1, 1, 1, 13, 7,
   4 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1, 19, 1, 1, 1, 1, 1,  1, 1,
   5 | 1, 1, 1,  1, 1, 1, 1,  1, 1, 19, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
   6 | 1, 1, 1, 17, 1, 1, 1,  1, 1,  1, 1, 17,  1,  1, 1, 1, 1, 1, 1, 17, 1,
   7 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1, 19,  1, 1, 1, 1, 1, 1, 29, 1,
   8 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
   9 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  10 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  11 | 1, 1, 1, 37, 1, 1, 1,  1, 1,  1, 1, 37,  1,  1, 1, 1, 1, 1, 1, 37, 1,
  12 | 1, 1, 1,  1, 1, 1, 1, 41, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  13 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  14 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  15 | 1, 1, 1,  1, 1, 1, 1,  1, 1, 61, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  16 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  17 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  18 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  19 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  20 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  21 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,

Cf. A000203, A003961, A246278, A342671, A355833.

Cf. also A355925, A355926, A355927 for similarly constructed arrays.

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));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
A355924sq(row,col) = A342671(A246278sq(row,col));
A355924list(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] = A355924sq(col,(a-(col-1))))); (v); };
v355924 = A355924list(up_to);
A355924(n) = v355924[n];

A(n,k) = A342671(A246278(n,k)).

A(n, k) = gcd(A246278(1+n,k), A355927(n, k)).

cp's OEIS Frontend

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

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A355925 Square array A(n, k) = A009194(A246278(n, k)), read by falling antidiagonals.

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A341607 Square array A(n,k) = A006530(A017666(A246278(n,k))), read by falling antidiagonals.

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A341626 Square array A(n,k) = A341526(A246278(n,k)), read by falling antidiagonals; Numerators of the columnwise first quotients of A341605/A341606.

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A378979 Square array A(n, k) = 2*A246278(n, k) - sigma(A246278(n, k)), read by falling antidiagonals. Deficiency applied to the prime shift array.

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A379010 Square array A(n, k) = phi(A246278(n, k)), read by falling antidiagonals; Euler totient function applied to the prime shift array.

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A246279 Transpose of array A246278: A(row,1) = 2*row, and for col > 1, A(row,col) = A003961(A(row,col-1)).

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A253551 Square array: A(row,col) = 2^(row-1) * 1+(2*A156552(col)) = A156552(A246278(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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