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A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

%I A372563 #11 May 21 2024 13:44:05
%S A372563 0,2,1,3,12,1,12,11,18,3,3,85,29,64,1,17,23,187,47,36,3,9,97,19,931,
%T A372563 53,106,1,50,17,291,75,733,71,54,3,36,504,35,889,31,2533,77,148,5,21,
%U A372563 121,1620,65,1011,111,1639,187,288,1,3,171,505,11840,59,2197,119,4927,179,90,5
%N A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.
%H A372563 Antti Karttunen, <a href="/A372563/b372563.txt">Table of n, a(n) for n = 1..11325; the first 150 antidiagonals, flattened</a>
%H A372563 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A372563 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A372563 A(n, k) = A286385(A246278(n, k)) = A246278(1+n, k) - A355927(n, k).
%e A372563 The top left corner of the array:
%e A372563 k=   1    2    3      4    5      6    7       8      9     10   11      12
%e A372563 2k=  2    4    6      8   10     12   14      16     18     20   22      24
%e A372563 ---+-------------------------------------------------------------------------
%e A372563 1  | 0,   2,   3,    12,   3,    17,   9,     50,    36,    21,   3,     75,
%e A372563 2  | 1,  12,  11,    85,  23,    97,  17,    504,   121,   171,  29,    635,
%e A372563 3  | 1,  18,  29,   187,  19,   291,  35,   1620,   505,   265,  25,   2525,
%e A372563 4  | 3,  64,  47,   931,  75,   889,  65,  11840,   795,  1259,  93,  12503,
%e A372563 5  | 1,  36,  53,   733,  31,  1011,  59,  12456,  1561,   817,  89,  16853,
%e A372563 6  | 3, 106,  71,  2533, 111,  2197, 157,  52580,  1839,  2987, 107,  50507,
%e A372563 7  | 1,  54,  77,  1639, 119,  2163,  49,  41580,  3193,  3101, 127,  53357,
%e A372563 8  | 3, 148, 187,  4927, 113,  6197, 211, 142280,  8283,  4969, 183, 179083,
%e A372563 9  | 5, 288, 179, 11669, 305,  9481, 277, 414720,  6965, 13421, 239, 374459,
%e A372563 10 | 1,  90, 187,  4531, 131,  7685,  73, 190980, 12649,  6303, 137, 293947,
%e A372563 11 | 5, 376, 301, 19869, 247, 18395, 331, 919856, 17173, 17161, 425, 906981,
%e A372563 12 | 3, 274, 167, 16861, 255, 13189, 349, 899540, 10335, 17099, 367, 777083,
%o A372563 (PARI)
%o A372563 up_to = 66;
%o A372563 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 A372563 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A372563 A286385(n) = (A003961(n)-sigma(n));
%o A372563 A372563sq(row,col) = A286385(A246278sq(row,col));
%o A372563 A372563list(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] = A372563sq(col,(a-(col-1))))); (v); };
%o A372563 v372563 = A372563list(up_to);
%o A372563 A372563(n) = v372563[n];
%Y A372563 Cf. A000203, A003961, A246278, A286385, A355927.
%Y A372563 Cf. A046933 (column 1).
%Y A372563 Cf. also A355924, A372562.
%K A372563 nonn,tabl
%O A372563 1,2
%A A372563 _Antti Karttunen_, May 21 2024