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