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

1,3

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.

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).

Formula

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

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

Example

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,

Prog

(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];

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.

Sequence in context: A349950 A164976 A261745 * A083714 A215481 A262933

Adjacent sequences: A378976 A378977 A378978 * A378980 A378981 A378982

Keyword

sign,tabl

Author

Antti Karttunen, Dec 13 2024

Status

approved