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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.

%I A372562 #17 May 21 2024 13:44:15
%S A372562 -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 A372562 645,-65,-49,-15,49,-1,189,5,-465,-119,-217,-15,39,463,-11,495,-127,
%U A372562 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 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.
%C A372562 For all k >= 1, A(1+A336836(2*k), k) < 0, and it is the topmost negative number of the column k.
%C A372562 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 A372562 Antti Karttunen, <a href="/A372562/b372562.txt">Table of n, a(n) for n = 1..11325; the first 150 antidiagonals, flattened</a>
%H A372562 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A372562 A(n,k) = A252748(A246278(n,k)).
%e A372562 The top left corner of the array:
%e A372562 k=    1     2     3      4     5      6     7       8      9     10    11      12
%e A372562 2k=   2     4     6      8    10     12    14      16     18     20    22      24
%e A372562 --+-------------------------------------------------------------------------------
%e A372562 1 |  -1,    1,    3,    11,    1,    21,    5,     49,    39,    23,   -5,     87,
%e A372562 2 |  -1,    7,    5,    71,   13,    85,   -1,    463,    95,   149,    7,    605,
%e A372562 3 |  -3,   -1,    7,    93,  -19,   189,  -11,   1151,   357,    87,  -37,   2023,
%e A372562 4 |  -3,   23,  -11,   645,    5,   495,  -29,   9839,   165,   783,  -13,   9757,
%e A372562 5 |  -9,  -73,  -65,  -465, -127,  -273, -119,   -721,    39,  -903, -129,   2743,
%e A372562 6 |  -9,  -49, -119,   519, -103,  -255, -105,  26399, -1377,   225, -227,  18649,
%e A372562 7 | -15, -217, -209, -2967, -231, -2679, -397, -36721, -2223, -2825, -351, -28937,
%e A372562 ...
%e A372562 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 A372562 (PARI)
%o A372562 up_to = 66;
%o A372562 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 A372562 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A372562 A252748(n) = (A003961(n) - (2*n));
%o A372562 A372562sq(row,col) = A252748(A246278sq(row,col));
%o A372562 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 A372562 v372562 = A372562list(up_to);
%o A372562 A372562(n) = v372562[n];
%Y A372562 Cf. A003961, A104210, A246278, A252748, A336836.
%Y A372562 Cf. A062234 (column 1 when values are negated).
%Y A372562 Cf. also A252750 (same terms in irregular triangle), A372563.
%Y A372562 See also conjecture 1 in A349753.
%K A372562 sign,tabl
%O A372562 1,4
%A A372562 _Antti Karttunen_, May 21 2024