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A344027 Arithmetic derivative applied to prime shift array: Square array A(n,k) = A003415(A246278(n,k)), read by falling antidiagonals.

%I A344027 #16 Jul 30 2022 08:22:56
%S A344027 1,4,1,5,6,1,12,8,10,1,7,27,12,14,1,16,10,75,18,22,1,9,39,16,147,24,
%T A344027 26,1,32,14,95,20,363,30,34,1,21,108,18,203,28,507,36,38,1,24,55,500,
%U A344027 24,407,32,867,42,46,1,13,51,119,1372,30,611,40,1083,52,58,1,44,16,135,275,5324,36,935,48,1587,60,62,1
%N A344027 Arithmetic derivative applied to prime shift array: Square array A(n,k) = A003415(A246278(n,k)), read by falling antidiagonals.
%C A344027 For each column k, A343221(2*k) gives the least n (row number) where A(n,k) < A246278(n,k).
%C A344027 Each column is monotonic.
%H A344027 Antti Karttunen, <a href="/A344027/b344027.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals</a>
%H A344027 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A344027 The top left corner of the array:
%e A344027     k = 1   2   3     4   5     6   7       8     9    10  11      12  13    14
%e A344027    2k = 2   4   6     8  10    12  14      16    18    20  22      24  26    28
%e A344027 ------+--------------------------------------------------------------------------
%e A344027   n=1 | 1,  4,  5,   12,  7,   16,  9,     32,   21,   24, 13,     44, 15,   32,
%e A344027     2 | 1,  6,  8,   27, 10,   39, 14,    108,   55,   51, 16,    162, 20,   75,
%e A344027     3 | 1, 10, 12,   75, 16,   95, 18,    500,  119,  135, 22,    650, 24,  155,
%e A344027     4 | 1, 14, 18,  147, 20,  203, 24,   1372,  275,  231, 26,   1960, 30,  287,
%e A344027     5 | 1, 22, 24,  363, 28,  407, 30,   5324,  455,  495, 34,   6050, 40,  539,
%e A344027     6 | 1, 26, 30,  507, 32,  611, 36,   8788,  731,  663, 42,  10816, 44,  767,
%e A344027     7 | 1, 34, 36,  867, 40,  935, 46,  19652, 1007, 1071, 48,  21386, 54, 1275,
%e A344027     8 | 1, 38, 42, 1083, 48, 1235, 50,  27436, 1403, 1463, 56,  31768, 60, 1539,
%e A344027     9 | 1, 46, 52, 1587, 54, 1863, 60,  48668, 2175, 1955, 64,  58190, 66, 2231,
%e A344027    10 | 1, 58, 60, 2523, 66, 2639, 70,  97556, 2759, 2987, 72, 102602, 76, 3219,
%e A344027    11 | 1, 62, 68, 2883, 72, 3255, 74, 119164, 3663, 3503, 78, 136462, 84, 3627,
%e A344027    12 | 1, 74, 78, 4107, 80, 4403, 84, 202612, 4715, 4551, 90, 219040, 96, 4847,
%e A344027 etc.
%o A344027 (PARI)
%o A344027 up_to = 91;
%o A344027 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A344027 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 A344027 A344027sq(row,col) = A003415(A246278sq(row,col));
%o A344027 A344027list(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] = A344027sq(col,(a-(col-1))))); (v); };
%o A344027 v344027 = A344027list(up_to);
%o A344027 A344027(n) = v344027[n];
%Y A344027 Cf. A003415, A246278.
%Y A344027 Cf. A068719 (row 1).
%Y A344027 Cf. also A343221, A343222, A344026, A355927, A356155.
%K A344027 nonn,tabl,look
%O A344027 1,2
%A A344027 _Antti Karttunen_, May 07 2021