A246278 -id:A246278 - cp's OEIS frontend

A372562 Square array A(n, k) = A246278(1+n, k) - 2*A246278(n, k), read by falling antidiagonals, where A246278 is the prime shift array.

-1, 1, -1, 3, 7, -3, 11, 5, -1, -3, 1, 71, 7, 23, -9, 21, 13, 93, -11, -73, -9, 5, 85, -19, 645, -65, -49, -15, 49, -1, 189, 5, -465, -119, -217, -15, 39, 463, -11, 495, -127, 519, -209, -193, -17, 23, 95, 1151, -29, -273, -103, -2967, -207, -217, -27, -5, 149, 357, 9839, -119, -255, -231, -1551, -435, -721, -25

Offset: 1

Antti Karttunen, May 21 2024

For all k >= 1, A(1+A336836(2*k), k) < 0, and it is the topmost negative number of the column k.

In those columns k where 2k is in A104210, 6, 12, 18, 24, ..., there is present a "prime thread" of successive primes (see the 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
--+-------------------------------------------------------------------------------
1 |  -1,    1,    3,    11,    1,    21,    5,     49,    39,    23,   -5,     87,
2 |  -1,    7,    5,    71,   13,    85,   -1,    463,    95,   149,    7,    605,
3 |  -3,   -1,    7,    93,  -19,   189,  -11,   1151,   357,    87,  -37,   2023,
4 |  -3,   23,  -11,   645,    5,   495,  -29,   9839,   165,   783,  -13,   9757,
5 |  -9,  -73,  -65,  -465, -127,  -273, -119,   -721,    39,  -903, -129,   2743,
6 |  -9,  -49, -119,   519, -103,  -255, -105,  26399, -1377,   225, -227,  18649,
7 | -15, -217, -209, -2967, -231, -2679, -397, -36721, -2223, -2825, -351, -28937,
...
Terms of column 9: 39 (3*13), 95 (5*19), 357 (3*7*17), 165 (3*5*11), 39 (3*13), -1377 (- 3^4 * 17), -2223 (- 3^2 * 13 * 19), ..., show an ascending "prime thread" (3, 5, 7, 11, 13, 17, 19, ...) that is mentioned in comments.

Cf. A003961, A104210, A246278, A252748, A336836.
Cf. A062234 (column 1 when values are negated).
Cf. also A252750 (same terms in irregular triangle), A372563.
See also conjecture 1 in A349753.

up_to = 66;
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); };
A252748(n) = (A003961(n) - (2*n));
A372562sq(row,col) = A252748(A246278sq(row,col));
A372562list(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] = A372562sq(col,(a-(col-1))))); (v); };
v372562 = A372562list(up_to);
A372562(n) = v372562[n];

A(n,k) = A252748(A246278(n,k)).

A253561 Square array read by antidiagonals: A(row,col) = A122111(A246278(row,col)).

Original entry on oeis.org

2, 3, 4, 6, 9, 8, 5, 18, 27, 16, 12, 25, 54, 81, 32, 10, 36, 125, 162, 243, 64, 24, 50, 108, 625, 486, 729, 128, 7, 72, 250, 324, 3125, 1458, 2187, 256, 15, 49, 216, 1250, 972, 15625, 4374, 6561, 512, 20, 75, 343, 648, 6250, 2916, 78125, 13122, 19683, 1024, 48, 100, 375, 2401, 1944, 31250, 8748, 390625, 39366, 59049, 2048, 14, 144, 500, 1875, 16807, 5832, 156250, 26244, 1953125, 118098, 177147, 4096

Offset: 2

Antti Karttunen, Jan 03 2015

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A253562 gives the inverse permutation.

The top row A253568 contains the same terms as A102750, but in different order.

			The top left corner of the array:
   2,  3,   6,   5,   12,   10,   24,    7,   15,   20,  48,   14,  96,   40,
   4,  9,  18,  25,   36,   50,   72,   49,   75,  100, 144,   98, 288,  200,
   8, 27,  54, 125,  108,  250,  216,  343,  375,  500, 432,  686, 864, 1000,
  16, 81, 162, 625,  324, 1250,  648, 2401, 1875, 2500,1296, 4802,2592, 5000,
  32,243, 486,3125,  972, 6250, 1944,16807, 9375,12500,3888,33614,7776,25000,
...

Antti Karttunen, Table of n, a(n) for n = 2..466; the first 30 antidiagonals of the array

Inverse: A253562.
The leftmost column: A000079. Topmost row: A253568.
Cf. A071178, A122111, A246278, A102750, A253560, A253563.

(define (A253561 n) (A122111 (A246278 n)))

a(n) = A122111(A246278(n)). [As a linear sequence].
Other identities.
A071178(A(row,col)) = row for all col. [All terms on row k have k as the exponent of their largest prime factor.]
A253560(A(row,col)) = A(row+1,col). [For any n >= 2, A253560(n) gives the term which is immediately below n in the same column of this array.]

A341628 Square array A(n,k) = A006530(A341527(A246278(n,k))), read by falling antidiagonals.

Original entry on oeis.org

3, 7, 5, 5, 13, 7, 3, 7, 31, 11, 7, 5, 11, 11, 13, 7, 11, 13, 13, 19, 17, 11, 13, 13, 11, 17, 61, 19, 31, 13, 31, 17, 61, 19, 307, 23, 13, 11, 17, 13, 19, 17, 23, 127, 29, 7, 31, 71, 19, 19, 23, 29, 29, 79, 31, 13, 13, 11, 2801, 23, 61, 29, 181, 31, 67, 37, 5, 17, 31, 19, 3221, 29, 307, 31, 53, 37, 331, 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
  2n=   2     4   6     8  10    12  14       16    18    20  22    24  26    28
-----+---------------------------------------------------------------------------
   1 |  3,    7,  5,    3,  7,    7, 11,      31,   13,    7, 13,    5, 17,   11,
   2 |  5,   13,  7,    5, 11,   13, 13,      11,   31,   13, 17,    7, 19,   13,
   3 |  7,   31, 11,   13, 13,   31, 17,      71,   11,   31, 19,   13, 23,   31,
   4 | 11,   11, 13,   11, 17,   13, 19,    2801,   19,   17, 23,   13, 29,   19,
   5 | 13,   19, 17,   61, 19,   19, 23,    3221,   61,   19, 29,   61, 31,   23,
   6 | 17,   61, 19,   17, 23,   61, 29,   30941,  307,   61, 31,   19, 37,   61,
   7 | 19,  307, 23,   29, 29,  307, 31,   88741,  127,  307, 37,   29, 41,  307,
   8 | 23,  127, 29,  181, 31,  127, 37,     911,   79,  127, 41,  181, 43,  127,
   9 | 29,   79, 31,   53, 37,   79, 41,  292561,   67,   79, 43,   53, 47,   79,
  10 | 31,   67, 37,  421, 41,   67, 43,  732541,  331,   67, 47,  421, 53,   67,
  11 | 37,  331, 41,   37, 43,  331, 47,   17351,   67,  331, 53,   41, 59,  331,
  12 | 41,   67, 43,  137, 47,   67, 53,    4271, 1723,   67, 59,  137, 61,   67,
  13 | 43, 1723, 47,   43, 53, 1723, 59,  579281,  631, 1723, 61,   47, 67, 1723,
  14 | 47,  631, 53,   47, 59,  631, 61, 3500201,   61,  631, 67,   53, 71,  631,
  15 | 53,   61, 59,   53, 61,   61, 67,   14621,  409,   61, 71,   59, 73,   67,
  16 | 59,  409, 61,  281, 67,  409, 71,    5581, 3541,  409, 73,  281, 79,  409,
  17 | 61, 3541, 67, 1741, 71, 3541, 73,     181,   97, 3541, 79, 1741, 83, 3541,
  18 | 67,   97, 71, 1861, 73,   97, 79,   21491,   71,   97, 83, 1861, 89,   97,
  19 | 71,   71, 73,  449, 79,   73, 83,   26881, 5113,   79, 89,  449, 97,   83,

Cf. A006530, A246278, A341527, A341627.

Cf. also A341607.

up_to = 105;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341527(n) = denominator(A341528(n) / A341529(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));
A341628sq(row,col) = A006530(A341527(A246278sq(row,col)));
A341628list(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] = A341628sq(col,(a-(col-1))))); (v); };
v341628 = A341628list(up_to);
A341628(n) = v341628[n];

A(n,k) = A006530(A341627(n,k)) = A006530(A341527(A246278(n,k))).

A250248 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),A078898(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 42, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 54, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 66, 67, 68, 135, 70, 71, 72, 103, 74, 51, 76, 77, 78, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3432
Index entries for sequences that are permutations of the natural numbers

Inverse: A250247.
Similar permutations: A250250 for even more recursed variant of A249818.
Cf. A000040, A005843, A049076, A049084, A055396, A078442, A078898, A246278.
Differs from the "vanilla version" A249818 for the first time at n=73, where a(73) = 108, while A249818(73) = 73.

a(1) = 1, a(n) = A246278(a(A055396(n)), A078898(n)).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n). [Fixes even numbers].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].

A252752 Inverse permutation to sequence A246278 when it is considered as a permutation of natural numbers (with assumption that a(1) = 1).

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 11, 8, 6, 12, 16, 17, 22, 23, 9, 30, 29, 38, 37, 47, 18, 57, 46, 68, 10, 80, 13, 93, 56, 107, 67, 122, 31, 138, 14, 155, 79, 173, 69, 192, 92, 212, 106, 233, 24, 255, 121, 278, 15, 302, 94, 327, 137, 353, 25, 380, 156, 408, 154, 437, 172, 467, 58, 498, 40, 530, 191, 563, 193, 597, 211, 632, 232, 668, 48, 705, 20

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse of array A246278 considered as a permutation of natural numbers with prepended a(1) = 1.
Cf. A055396, A246277.
Related permutations A122111, A156552, A246276, A253552, A253562.
Differs from A252460 for the first time at n=21, where a(21) = 18, while A252460(21) = 13.

a(1) = 1; for n>1: a(n) = 1 + A246276(n-1).
As a composition of related permutations:
a(n) = A253562(A122111(n)).
a(n) = 1 + A253552(A156552(n)).

A379011 Square array A(n, k) = 2phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2phi(n)-n), applied to the prime shift array.

Original entry on oeis.org

0, 0, 1, -2, 3, 3, 0, 1, 15, 5, -2, 9, 13, 35, 9, -4, 3, 75, 43, 99, 11, -2, 3, 25, 245, 97, 143, 15, 0, 7, 65, 53, 1089, 163, 255, 17, -6, 27, 31, 301, 133, 1859, 253, 323, 21, -4, 5, 375, 73, 1067, 185, 4335, 355, 483, 27, -2, 9, 91, 1715, 151, 2119, 313, 6137, 565, 783, 29, -8, 9, 125, 473, 11979, 229, 4301, 457, 11109, 781, 899, 35

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   11      12
2k= |  2    4    6      8   10     12   14      16     18     20   22      24
----+-------------------------------------------------------------------------
  1 |  0,   0,  -2,     0,  -2,    -4,  -2,      0,    -6,    -4,  -2,     -8,
  2 |  1,   3,   1,     9,   3,     3,   7,     27,     5,     9,   9,      9,
  3 |  3,  15,  13,    75,  25,    65,  31,    375,    91,   125,  43,    325,
  4 |  5,  35,  43,   245,  53,   301,  73,   1715,   473,   371,  83,   2107,
  5 |  9,  99,  97,  1089, 133,  1067, 151,  11979,  1261,  1463, 187,  11737,
  6 | 11, 143, 163,  1859, 185,  2119, 229,  24167,  2771,  2405, 295,  27547,
  7 | 15, 255, 253,  4335, 313,  4301, 403,  73695,  4807,  5321, 433,  73117,
  8 | 17, 323, 355,  6137, 457,  6745, 491, 116603,  8165,  8683, 593, 128155,
  9 | 21, 483, 565, 11109, 607, 12995, 733, 255507, 16385, 13961, 817, 298885,

Cf. A000010, A083254, A246278, A379010.
Cf. A040976 (column 1), A378986 (row 1).
Cf. also A378979.

up_to = 11325; \\ = binomial(150+1,2)
A083254(n) = (2*eulerphi(n)-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));
A379011sq(row,col) = A083254(A246278sq(row,col));
A379011list(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] = A379011sq(col,(a-(col-1))))); (v); };
v379011 = A379011list(up_to);
A379011(n) = v379011[n];

A(n, k) = 2*A379010(n, k) - A246278(n, k).

A355926 Square array A(n,k) = A355442(A246278(n,k)), read by falling antidiagonals.

Original entry on oeis.org

3, 9, 1, 5, 5, 1, 3, 5, 1, 1, 3, 125, 7, 1, 1, 5, 5, 343, 1, 1, 1, 3, 175, 7, 11, 1, 1, 1, 9, 1, 49, 1, 1, 1, 1, 1, 25, 125, 7, 121, 1, 1, 1, 1, 1, 3, 245, 2401, 1, 1, 1, 1, 1, 1, 1, 3, 1, 77, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 49, 11, 28561, 1, 1, 1, 1, 1, 1, 1, 3, 175, 7, 121, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 5, 77, 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:
   n= 1  2  3    4  5    6  7      8    9   10 11     12 13   14  15       16
  2n= 2  4  6    8 10   12 14     16   18   20 22     24 26   28  30       32
----+--------------------------------------------------------------------------
  1 | 3, 9, 5,   3, 3,   5, 3,     9,  25,   3, 3,     5, 3,   9,  7,       3,
  2 | 1, 5, 5, 125, 5, 175, 1,   125, 245,   1, 5,   175, 5,   5, 35,       1,
  3 | 1, 1, 7, 343, 7,  49, 7,  2401,  77,  49, 7,    77, 7,  49, 77,   16807,
  4 | 1, 1, 1,  11, 1, 121, 1,     1,  11, 121, 1, 17303, 1, 121, 11,    1331,
  5 | 1, 1, 1,   1, 1,   1, 1, 28561,   1,   1, 1,  2197, 1,   1, 13,   28561,
  6 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,    17, 1,   1,  1, 1419857,
  7 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,     1, 1,   1,  1,     361,
  8 | 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,

Cf. A003961, A246278, A276086, A355442, A355835.

Cf. also A355924, A355925 for similarly constructed arrays.

up_to = 105;
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(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));
A355926sq(row,col) = A355442(A246278sq(row,col));
A355926list(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] = A355926sq(col,(a-(col-1))))); (v); };
v355926 = A355926list(up_to);
A355926(n) = v355926[n];

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

Original entry on oeis.org

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

Antti Karttunen, Jul 29 2022

Each column is strictly monotonic.

			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,

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

Cf. also A344027.

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];

A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

Original entry on oeis.org

0, 2, 1, 3, 12, 1, 12, 11, 18, 3, 3, 85, 29, 64, 1, 17, 23, 187, 47, 36, 3, 9, 97, 19, 931, 53, 106, 1, 50, 17, 291, 75, 733, 71, 54, 3, 36, 504, 35, 889, 31, 2533, 77, 148, 5, 21, 121, 1620, 65, 1011, 111, 1639, 187, 288, 1, 3, 171, 505, 11840, 59, 2197, 119, 4927, 179, 90, 5

Offset: 1

Antti Karttunen, May 21 2024

			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  | 0,   2,   3,    12,   3,    17,   9,     50,    36,    21,   3,     75,
2  | 1,  12,  11,    85,  23,    97,  17,    504,   121,   171,  29,    635,
3  | 1,  18,  29,   187,  19,   291,  35,   1620,   505,   265,  25,   2525,
4  | 3,  64,  47,   931,  75,   889,  65,  11840,   795,  1259,  93,  12503,
5  | 1,  36,  53,   733,  31,  1011,  59,  12456,  1561,   817,  89,  16853,
6  | 3, 106,  71,  2533, 111,  2197, 157,  52580,  1839,  2987, 107,  50507,
7  | 1,  54,  77,  1639, 119,  2163,  49,  41580,  3193,  3101, 127,  53357,
8  | 3, 148, 187,  4927, 113,  6197, 211, 142280,  8283,  4969, 183, 179083,
9  | 5, 288, 179, 11669, 305,  9481, 277, 414720,  6965, 13421, 239, 374459,
10 | 1,  90, 187,  4531, 131,  7685,  73, 190980, 12649,  6303, 137, 293947,
11 | 5, 376, 301, 19869, 247, 18395, 331, 919856, 17173, 17161, 425, 906981,
12 | 3, 274, 167, 16861, 255, 13189, 349, 899540, 10335, 17099, 367, 777083,

Cf. A000203, A003961, A246278, A286385, A355927.
Cf. A046933 (column 1).
Cf. also A355924, A372562.

up_to = 66;
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); };
A286385(n) = (A003961(n)-sigma(n));
A372563sq(row,col) = A286385(A246278sq(row,col));
A372563list(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] = A372563sq(col,(a-(col-1))))); (v); };
v372563 = A372563list(up_to);
A372563(n) = v372563[n];

A(n, k) = A286385(A246278(n, k)) = A246278(1+n, k) - A355927(n, k).

A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

Original entry on oeis.org

0, 0, 0, -2, 3, 2, 0, 2, 16, 3, 0, 10, 19, 38, 7, -6, 7, 88, 54, 104, 9, 1, 8, 33, 280, 113, 151, 14, 0, 16, 96, 65, 1192, 184, 268, 15, -5, 38, 44, 389, 152, 2009, 282, 336, 18, -4, 22, 464, 88, 1279, 207, 4600, 388, 502, 24, 5, 16, 142, 1996, 174, 2445, 345, 6470, 608, 806, 25, -14, 18, 174, 623, 13170, 257, 4834, 497, 11605, 833, 924, 33

Offset: 1

Antti Karttunen, Dec 14 2024

Question: Are all columns increasing, and strictly increasing after the leftmost column?

			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   |  0,   0,  -2,     0,   0,    -6,   1,      0,    -5,    -4,   5,    -14,
2   |  0,   3,   2,    10,   7,     8,  16,     38,    22,    16,  18,     26,
3   |  2,  16,  19,    88,  33,    96,  44,    464,   142,   174,  58,    495,
4   |  3,  38,  54,   280,  65,   389,  88,   1996,   623,   469, 103,   2737,
5   |  7, 104, 113,  1192, 152,  1279, 174,  13170,  1516,  1717, 211,  14102,
6   |  9, 151, 184,  2009, 207,  2445, 257,  26172,  3208,  2756, 328,  31850,
7   | 14, 268, 282,  4600, 345,  4834, 439,  78295,  5406,  5916, 473,  82285,
8   | 15, 336, 388,  6470, 497,  7455, 533, 123071,  9035,  9501, 638, 141745,
9   | 18, 502, 608, 11605, 653, 14081, 784, 267115, 17773, 15097, 870, 324077,
Here 0's occur also after the first row. For example column 30, which corresponds with numbers 60, 315, 1925, 7007, 26741, ..., begins as -52, 0, 868, 4428, 19958, etc. See also A295296.

Cf. A000203, A005187, A246278, A294898, A295296.
Cf. A080085 (column 1, incremented by one).
Cf. also array A378979, and A324348 (another permutation of A294898).

up_to = 11325; \\ = binomial(150+1,2)
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(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));
A379008sq(row,col) = A294898(A246278sq(row,col));
A379008list(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] = A379008sq(col,(a-(col-1))))); (v); };
v379008 = A379008list(up_to);
A379008(n) = v379008[n];

cp's OEIS Frontend

A372562 Square array A(n, k) = A246278(1+n, k) - 2*A246278(n, k), read by falling antidiagonals, where A246278 is the prime shift array.

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A253561 Square array read by antidiagonals: A(row,col) = A122111(A246278(row,col)).

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A341628 Square array A(n,k) = A006530(A341527(A246278(n,k))), read by falling antidiagonals.

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A250248 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),A078898(n)).

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A252752 Inverse permutation to sequence A246278 when it is considered as a permutation of natural numbers (with assumption that a(1) = 1).

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A379011 Square array A(n, k) = 2*phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2*phi(n)-n), applied to the prime shift array.

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A355926 Square array A(n,k) = A355442(A246278(n,k)), read by falling antidiagonals.

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

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A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

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A379011 Square array A(n, k) = 2phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2phi(n)-n), applied to the prime shift array.