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A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

%I A379008 #11 Dec 14 2024 14:42:08
%S A379008 0,0,0,-2,3,2,0,2,16,3,0,10,19,38,7,-6,7,88,54,104,9,1,8,33,280,113,
%T A379008 151,14,0,16,96,65,1192,184,268,15,-5,38,44,389,152,2009,282,336,18,
%U A379008 -4,22,464,88,1279,207,4600,388,502,24,5,16,142,1996,174,2445,345,6470,608,806,25,-14,18,174,623,13170,257,4834,497,11605,833,924,33
%N A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.
%C A379008 Question: Are all columns increasing, and strictly increasing after the leftmost column?
%H A379008 Antti Karttunen, <a href="/A379008/b379008.txt">Table of n, a(n) for n = 1..11325; the first 150 antidiagonals, flattened</a>
%H A379008 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%H A379008 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%e A379008 The top left corner of the array:
%e A379008 k=  |  1    2    3      4    5      6    7       8      9     10   11      12
%e A379008 2k= |  2    4    6      8   10     12   14      16     18     20   22      24
%e A379008 ----+-------------------------------------------------------------------------
%e A379008 1   |  0,   0,  -2,     0,   0,    -6,   1,      0,    -5,    -4,   5,    -14,
%e A379008 2   |  0,   3,   2,    10,   7,     8,  16,     38,    22,    16,  18,     26,
%e A379008 3   |  2,  16,  19,    88,  33,    96,  44,    464,   142,   174,  58,    495,
%e A379008 4   |  3,  38,  54,   280,  65,   389,  88,   1996,   623,   469, 103,   2737,
%e A379008 5   |  7, 104, 113,  1192, 152,  1279, 174,  13170,  1516,  1717, 211,  14102,
%e A379008 6   |  9, 151, 184,  2009, 207,  2445, 257,  26172,  3208,  2756, 328,  31850,
%e A379008 7   | 14, 268, 282,  4600, 345,  4834, 439,  78295,  5406,  5916, 473,  82285,
%e A379008 8   | 15, 336, 388,  6470, 497,  7455, 533, 123071,  9035,  9501, 638, 141745,
%e A379008 9   | 18, 502, 608, 11605, 653, 14081, 784, 267115, 17773, 15097, 870, 324077,
%e A379008 Here 0's occur also after the first row. For example column 30, which corresponds with numbers 60, 315, 1925, 7007, 26741, ..., begins as -52, 0, 868, 4428, 19958, etc. See also A295296.
%o A379008 (PARI)
%o A379008 up_to = 11325; \\ = binomial(150+1,2)
%o A379008 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A379008 A294898(n) = (A005187(n)-sigma(n));
%o A379008 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 A379008 A379008sq(row,col) = A294898(A246278sq(row,col));
%o A379008 A379008list(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] = A379008sq(col,(a-(col-1))))); (v); };
%o A379008 v379008 = A379008list(up_to);
%o A379008 A379008(n) = v379008[n];
%Y A379008 Cf. A000203, A005187, A246278, A294898, A295296.
%Y A379008 Cf. A080085 (column 1, incremented by one).
%Y A379008 Cf. also array A378979, and A324348 (another permutation of A294898).
%K A379008 sign,tabl
%O A379008 1,4
%A A379008 _Antti Karttunen_, Dec 14 2024