A372562 - OEIS

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

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OFFSET

1,4

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).

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

FORMULA

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

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.

PROG

(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];

CROSSREFS

Cf. A003961, A104210, A246278, A252748, A336836.

Cf. A062234 (column 1 when values are negated).

Cf. also A252750 (same terms in irregular triangle), A372563.