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%I #8 Feb 20 2021 07:53:09
%S 8,52,9,4,279,20,64,6,1425,21,160,1053,10,343,77,26,189,12500,49,
%T 22143,117,28,372,110,62769,33,51883,170,1936,231,4275,351,791945,130,
%U 110109,114,248,5751,780,2401,6545,573417,68,199633,115,1040,2565,1750625,595,199287,13338,1778506,57,460759,464
%N Square array A(n,k) = A341526(A246278(n,k)), read by falling antidiagonals; Numerators of the columnwise first quotients of A341605/A341606.
%C See comments in A341605.
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A(n,k) = A341526(A246278(n,k)).
%F If we set r(row,col) = A341605(row,col)/A341606(row,col) and d(row,col) = A(row,col)/A341627(row,col), then d(row,col) = r(row+1,col)/r(row,col).
%F For all n, k, A(n,k) < A341627(n, k).
%e The top left corner of the array:
%e n = 1 2 3 4 5 6 7 8 9
%e 2n = 2 4 6 8 10 12 14 16 18
%e ----+--------------------------------------------------------------------------
%e 1 | 8, 52, 4, 64, 160, 26, 28, 1936, 248,
%e 2 | 9, 279, 6, 1053, 189, 372, 231, 5751, 2565,
%e 3 | 20, 1425, 10, 12500, 110, 4275, 780, 1750625, 980,
%e 4 | 21, 343, 49, 62769, 351, 2401, 595, 38668105, 6039,
%e 5 | 77, 22143, 33, 791945, 6545, 199287, 1463, 453007181, 307307,
%e 6 | 117, 51883, 130, 573417, 13338, 518830, 13455, 2534531701, 757809,
%e 7 | 170, 110109, 68, 1778506, 9775, 660654, 15776, 11489232281, 1786190,
%e 8 | 114, 199633, 57, 2181162, 17632, 998165, 33573, 38126842081, 2283762,
%e 9 | 115, 460759, 92, 5122307, 67735, 7372144, 89355, 204995005981, 3311655,
%e etc.
%o (PARI)
%o up_to = 105;
%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 A341626sq(row,col) = A341526(A246278sq(row,col));
%o A341626list(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] = A341626sq(col,(a-(col-1))))); (v); };
%o v341626 = A341626list(up_to);
%o A341626(n) = v341626[n];
%Y Cf. A246278, A341605, A341606, A341526.
%Y Cf. A341627 (denominators).
%K nonn,frac,tabl
%O 1,1
%A _Antti Karttunen_, Feb 16 2021
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