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%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
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