login
Square array A(n, k) = A246278(1+n, k) - 2*A246278(n, k), read by falling antidiagonals, where A246278 is the prime shift array.
4

%I #17 May 21 2024 13:44:15

%S -1,1,-1,3,7,-3,11,5,-1,-3,1,71,7,23,-9,21,13,93,-11,-73,-9,5,85,-19,

%T 645,-65,-49,-15,49,-1,189,5,-465,-119,-217,-15,39,463,-11,495,-127,

%U 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

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

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

%C 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).

%H Antti Karttunen, <a href="/A372562/b372562.txt">Table of n, a(n) for n = 1..11325; the first 150 antidiagonals, flattened</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

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

%e The top left corner of the array:

%e k= 1 2 3 4 5 6 7 8 9 10 11 12

%e 2k= 2 4 6 8 10 12 14 16 18 20 22 24

%e --+-------------------------------------------------------------------------------

%e 1 | -1, 1, 3, 11, 1, 21, 5, 49, 39, 23, -5, 87,

%e 2 | -1, 7, 5, 71, 13, 85, -1, 463, 95, 149, 7, 605,

%e 3 | -3, -1, 7, 93, -19, 189, -11, 1151, 357, 87, -37, 2023,

%e 4 | -3, 23, -11, 645, 5, 495, -29, 9839, 165, 783, -13, 9757,

%e 5 | -9, -73, -65, -465, -127, -273, -119, -721, 39, -903, -129, 2743,

%e 6 | -9, -49, -119, 519, -103, -255, -105, 26399, -1377, 225, -227, 18649,

%e 7 | -15, -217, -209, -2967, -231, -2679, -397, -36721, -2223, -2825, -351, -28937,

%e ...

%e 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.

%o (PARI)

%o up_to = 66;

%o 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));

%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A252748(n) = (A003961(n) - (2*n));

%o A372562sq(row,col) = A252748(A246278sq(row,col));

%o 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); };

%o v372562 = A372562list(up_to);

%o A372562(n) = v372562[n];

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

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

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

%Y See also conjecture 1 in A349753.

%K sign,tabl

%O 1,4

%A _Antti Karttunen_, May 21 2024