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

Author

Antti Karttunen, May 21 2024

Keywords

• sign
• tabl

Comments

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).

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..11325; the first 150 antidiagonals, flattened
• Index entries for sequences computed from indices in prime factorization

Crossrefs

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.

Programs

PARI

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];

Formula

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