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A341626 Square array A(n,k) = A341526(A246278(n,k)), read by falling antidiagonals; Numerators of the columnwise first quotients of A341605/A341606.

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