A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

0, 0, 0, -2, 3, 2, 0, 2, 16, 3, 0, 10, 19, 38, 7, -6, 7, 88, 54, 104, 9, 1, 8, 33, 280, 113, 151, 14, 0, 16, 96, 65, 1192, 184, 268, 15, -5, 38, 44, 389, 152, 2009, 282, 336, 18, -4, 22, 464, 88, 1279, 207, 4600, 388, 502, 24, 5, 16, 142, 1996, 174, 2445, 345, 6470, 608, 806, 25, -14, 18, 174, 623, 13170, 257, 4834, 497, 11605, 833, 924, 33

Offset

1,4

Comments

Question: Are all columns increasing, and strictly increasing after the leftmost column?

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).

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   |  0,   0,  -2,     0,   0,    -6,   1,      0,    -5,    -4,   5,    -14,
2   |  0,   3,   2,    10,   7,     8,  16,     38,    22,    16,  18,     26,
3   |  2,  16,  19,    88,  33,    96,  44,    464,   142,   174,  58,    495,
4   |  3,  38,  54,   280,  65,   389,  88,   1996,   623,   469, 103,   2737,
5   |  7, 104, 113,  1192, 152,  1279, 174,  13170,  1516,  1717, 211,  14102,
6   |  9, 151, 184,  2009, 207,  2445, 257,  26172,  3208,  2756, 328,  31850,
7   | 14, 268, 282,  4600, 345,  4834, 439,  78295,  5406,  5916, 473,  82285,
8   | 15, 336, 388,  6470, 497,  7455, 533, 123071,  9035,  9501, 638, 141745,
9   | 18, 502, 608, 11605, 653, 14081, 784, 267115, 17773, 15097, 870, 324077,
Here 0's occur also after the first row. For example column 30, which corresponds with numbers 60, 315, 1925, 7007, 26741, ..., begins as -52, 0, 868, 4428, 19958, etc. See also A295296.

Prog

(PARI)
up_to = 11325; \\ = binomial(150+1, 2)
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(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));
A379008sq(row, col) = A294898(A246278sq(row, col));
A379008list(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] = A379008sq(col, (a-(col-1))))); (v); };
v379008 = A379008list(up_to);
A379008(n) = v379008[n];

Crossrefs

Cf. A000203, A005187, A246278, A294898, A295296.

Cf. A080085 (column 1, incremented by one).

Cf. also array A378979, and A324348 (another permutation of A294898).

Sequence in context: A056619 A324300 A323695 * A303121 A332921 A239579

Adjacent sequences: A379005 A379006 A379007 * A379009 A379010 A379011

Keyword

sign,tabl

Author

Antti Karttunen, Dec 14 2024

Status

approved