A372563 -id:A372563 - cp's OEIS frontend

A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

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

Offset: 2

Antti Karttunen, Aug 21 2014

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
This array can be obtained by taking every second column from array A242378, starting from its column 2.
Permutation of natural numbers larger than 1.
The terms on row n are all divisible by n-th prime, A000040(n).
Each column is strictly growing, and the terms in the same column have the same prime signature.
A055396(n) gives the row number of row where n occurs,
and A246277(n) gives its column number, both starting from 1.
From Antti Karttunen, Jan 03 2015: (Start)
A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.
(End)

			The top left corner of the array:
   2,     4,     6,     8,    10,    12,    14,    16,    18, ...
   3,     9,    15,    27,    21,    45,    33,    81,    75, ...
   5,    25,    35,   125,    55,   175,    65,   625,   245, ...
   7,    49,    77,   343,    91,   539,   119,  2401,   847, ...
  11,   121,   143,  1331,   187,  1573,   209, 14641,  1859, ...
  13,   169,   221,  2197,   247,  2873,   299, 28561,  3757, ...

Antti Karttunen, Table of n, a(n) for n = 2..1276; the first 50 antidiagonals of the array

First row: A005843 (the even numbers), from 2 onward.
Row 2: A249734, Row 3: A249827.
Column 1: A000040 (primes), Column 2: A001248 (squares of primes), Column 3: A006094 (products of two successive primes), Column 4: A030078 (cubes of primes).
Transpose: A246279.
Inverse permutation: A252752.
One more than A246275.
Cf. A005940, A242378, A246259, A000040, A002260, A004736, A003961, A055396, A083221, A114537, A246277 (terms of A348717 halved), A246675, A246684, A249818, A252759, A253515.
Arrays obtained by applying a particular function (given in parentheses) to the entries of this array. Cases where the columns grow monotonically are indicated with *: A249822 (A078898), A253551 (* A156552), A253561 (* A122111), A341605 (A017665), A341606 (A017666), A341607 (A006530 o A017666), A341608 (A341524), A341626 (A341526), A341627 (A341527), A341628 (A006530 o A341527), A342674 (A341530), A344027 (* A003415, arithmetic derivative), A355924 (A342671), A355925 (A009194), A355926 (A355442), A355927 (* sigma), A356155 (* A258851), A372562 (A252748), A372563 (A286385), A378979 (* deficiency, A033879), A379008 (* (probably), A294898), A379010 (* A000010, Euler phi), A379011 (* A083254).
Cf. A329050 (subtable).

f[p_?PrimeQ] := f[p] = Prime[PrimePi@ p + 1]; f[1] = 1; f[n_] := f[n] = Times @@ (f[First@ #]^Last@ # &) /@ FactorInteger@ n; Block[{lim = 12}, Table[#[[n - k, k]], {n, 2, lim}, {k, n - 1, 1, -1}] &@ NestList[Map[f, #] &, Table[2 k, {k, lim}], lim]] // Flatten (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 *)

(define (A246278 n) (if (<= n 1) n (A246278bi (A002260 (- n 1)) (A004736 (- n 1))))) ;; Square array starts with offset=2, and we have also tacitly defined a(1) = 1 here.
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
As a composition of other similar sequences:
a(n) = A122111(A253561(n)).
a(n) = A249818(A083221(n)).
For all n >= 1, a(n+1) = A005940(1+A253551(n)).
A(n, k) = A341606(n, k) * A355925(n, k). - Antti Karttunen, Jul 22 2022

Starting offset of the linear sequence changed from 1 to 2, without affecting the column and row indices by Antti Karttunen, Jan 03 2015

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).

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.

Original entry on oeis.org

-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)).

cp's OEIS Frontend

A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

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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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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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