A378979 Square array A(n, k) = 2*A246278(n, k) - sigma(A246278(n, k)), read by falling antidiagonals. Deficiency applied to the prime shift array.

1, 1, 2, 0, 5, 4, 1, 6, 19, 6, 2, 14, 22, 41, 10, -4, 10, 94, 58, 109, 12, 4, 12, 38, 286, 118, 155, 16, 1, 18, 102, 70, 1198, 190, 271, 18, -3, 41, 46, 394, 158, 2014, 286, 341, 22, -2, 26, 469, 94, 1284, 214, 4606, 394, 505, 28, 8, 22, 148, 2001, 178, 2452, 350, 6478, 614, 811, 30, -12, 22, 178, 630, 13177, 262, 4842, 502, 11614, 838, 929, 36

Offset: 1

Author

Antti Karttunen, Dec 13 2024

Keywords

• sign
• tabl

Comments

Each column is strictly increasing.

For all k >= 1, A(1+A378985(k), k) > 0, and it is the topmost positive number of the column k.

Examples

			The top left corner of the array:
k=  |  1    2    3     4    5     6    7      8     9    10   11     12
2k= |  2    4    6     8   10    12   14     16    18    20   22     24
----+-------------------------------------------------------------------
  1 |  1,   1,   0,    1,   2,   -4,   4,     1,   -3,   -2,   8,   -12,
  2 |  2,   5,   6,   14,  10,   12,  18,    41,   26,   22,  22,    30,
  3 |  4,  19,  22,   94,  38,  102,  46,   469,  148,  178,  62,   502,
  4 |  6,  41,  58,  286,  70,  394,  94,  2001,  630,  476, 106,  2746,
  5 | 10, 109, 118, 1198, 158, 1284, 178, 13177, 1522, 1720, 218, 14110,
  6 | 12, 155, 190, 2014, 214, 2452, 262, 26181, 3216, 2762, 334, 31858,
  7 | 16, 271, 286, 4606, 350, 4842, 446, 78301, 5416, 5926, 478, 82294,
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..11325; the first 150 antidiagonals, flattened
• Index entries for sequences related to prime indices in the factorization of n.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A033879, A246278, A336835, A355927, A372562, A372563.

Cf. A006093 (column 1), A306190 (column 2), A378978 (row 1), A378985 (row index of the topmost positive term in column n).

Cf. also arrays A341605, A341606 and A341607.

Cf. also A324055.

Programs

PARI

up_to = 78;
A033879(n) = (n+n-sigma(n));
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));
A378979sq(row,col) = A033879(A246278sq(row,col));
A378979list(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] = A378979sq(col,(a-(col-1))))); (v); };
v378979 = A378979list(up_to);
A378979(n) = v378979[n];

Formula

A(n, k) = A033879(A246278(n, k)) = 2*A246278(n, k) - A355927(n, k).

A(n, k) = A372563(n,k) - A372562(n, k).