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A378979 Square array A(n, k) = 2*A246278(n, k) - sigma(A246278(n, k)), read by falling antidiagonals. Deficiency applied to the prime shift array.

%I A378979 #23 Dec 15 2024 06:49:49
%S A378979 1,1,2,0,5,4,1,6,19,6,2,14,22,41,10,-4,10,94,58,109,12,4,12,38,286,
%T A378979 118,155,16,1,18,102,70,1198,190,271,18,-3,41,46,394,158,2014,286,341,
%U A378979 22,-2,26,469,94,1284,214,4606,394,505,28,8,22,148,2001,178,2452,350,6478,614,811,30,-12,22,178,630,13177,262,4842,502,11614,838,929,36
%N A378979 Square array A(n, k) = 2*A246278(n, k) - sigma(A246278(n, k)), read by falling antidiagonals. Deficiency applied to the prime shift array.
%C A378979 Each column is strictly increasing.
%C A378979 For all k >= 1, A(1+A378985(k), k) > 0, and it is the topmost positive number of the column k.
%H A378979 Antti Karttunen, <a href="/A378979/b378979.txt">Table of n, a(n) for n = 1..11325; the first 150 antidiagonals, flattened</a>
%H A378979 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%H A378979 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A378979 A(n, k) = A033879(A246278(n, k)) = 2*A246278(n, k) - A355927(n, k).
%F A378979 A(n, k) = A372563(n,k) - A372562(n, k).
%e A378979 The top left corner of the array:
%e A378979 k=  |  1    2    3     4    5     6    7      8     9    10   11     12
%e A378979 2k= |  2    4    6     8   10    12   14     16    18    20   22     24
%e A378979 ----+-------------------------------------------------------------------
%e A378979   1 |  1,   1,   0,    1,   2,   -4,   4,     1,   -3,   -2,   8,   -12,
%e A378979   2 |  2,   5,   6,   14,  10,   12,  18,    41,   26,   22,  22,    30,
%e A378979   3 |  4,  19,  22,   94,  38,  102,  46,   469,  148,  178,  62,   502,
%e A378979   4 |  6,  41,  58,  286,  70,  394,  94,  2001,  630,  476, 106,  2746,
%e A378979   5 | 10, 109, 118, 1198, 158, 1284, 178, 13177, 1522, 1720, 218, 14110,
%e A378979   6 | 12, 155, 190, 2014, 214, 2452, 262, 26181, 3216, 2762, 334, 31858,
%e A378979   7 | 16, 271, 286, 4606, 350, 4842, 446, 78301, 5416, 5926, 478, 82294,
%o A378979 (PARI)
%o A378979 up_to = 78;
%o A378979 A033879(n) = (n+n-sigma(n));
%o A378979 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 A378979 A378979sq(row,col) = A033879(A246278sq(row,col));
%o A378979 A378979list(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] = A378979sq(col,(a-(col-1))))); (v); };
%o A378979 v378979 = A378979list(up_to);
%o A378979 A378979(n) = v378979[n];
%Y A378979 Cf. A033879, A246278, A336835, A355927, A372562, A372563.
%Y A378979 Cf. A006093 (column 1), A306190 (column 2), A378978 (row 1), A378985 (row index of the topmost positive term in column n).
%Y A378979 Cf. also arrays A341605, A341606 and A341607.
%Y A378979 Cf. also A324055.
%K A378979 sign,tabl
%O A378979 1,3
%A A378979 _Antti Karttunen_, Dec 13 2024