A394890 Irregular triangle T(n, k) = number of integers x in range A002110(n-1) .. A002110(n)-1 such that the u-th arithmetic derivative of A276086(x) is zero for some u, and A328114(x) = k, where A002110(n) is the n-th primorial, and A328114 gives the largest digit in the primorial base.

1, 2, 2, 3, 2, 2, 1, 6, 14, 9, 9, 2, 4, 13, 34, 31, 46, 26, 39, 12, 6, 4, 5, 18, 101, 159, 274, 232, 221, 133, 108, 95, 79, 31, 23, 36, 262, 639, 1225, 1385, 1654, 1239, 1166, 1051, 959, 568, 420, 189, 133, 128, 76, 72, 755, 2371, 5548, 7881, 10369, 9863, 10143, 10225, 10140, 6977, 6348, 3564, 2878, 2217, 1859, 725, 554

Offset

1,2

Comments

T(n, k) is the number of terms of A099308 (or A351255) whose greatest prime factor (A006530) is prime(n) [in other words, that are prime(n)-smooth but not prime(n)-smooth], and in whose prime factorization the maximum exponent (A051903) is k, where k ranges from 1 to prime(n)-1.

Links

Table of n, a(n) for n=1..69.

Example

Beginning of the irregular triangle. Each row n has prime(n)-1 elements:
n\k|  1    2     3     4     5      6     7      8     9      10    11    12   ...
---+-------------------------------------------------------------------------------
1  |  1,
2  |  2,   2,
3  |  3,   2,    2,    1,
4  |  6,  14,    9,    9,    2,     4,
5  | 13,  34,   31,   46,   26,    39,   12,     6,     4,     5,
6  | 18, 101,  159,  274,  232,   221,  133,   108,    95,    79,   31,   23,
7  | 36, 262,  639, 1225, 1385,  1654, 1239,  1166,  1051,   959,  568,  420,  189,  133,  128,   76,
8  | 72, 755, 2371, 5548, 7881, 10369, 9863, 10143, 10225, 10140, 6977, 6348, 3564, 2878, 2217, 1859, 725, 554,

Prog

(PARI)
v328116 = readvec("a328116.txt"); \\ Use the data file of 105368 terms given in A328116
up_to_n_rows = 8;
A002110(n) = prod(i=1, n, prime(i));
A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };
A394890_sq(n, k) = { my(si=vecsearch(v328116, A002110(n-1)), ei=vecsearch(v328116, A002110(n))); sum(i=si, ei-1, k==A328114(v328116[i])); };
A394890_list(up_to_row) = { my(lista=List([])); for(row=1, up_to_row, for(k=1, prime(row)-1, listput(lista, A394890_sq(row, k)))); Vec(lista); };
v394890 = A394890_list(up_to_n_rows);
A394890(n) = v394890[n];

Crossrefs

Cf. A003415, A006530, A002110, A051903, A099308, A276086, A328114, A328116, A351255.

Cf. A006093 (row lengths), A351071 (row sums), A394893 (the right edge), A394894 (the left edge).

Sequence in context: A251140 A259977 A341434 * A115312 A237721 A254296

Adjacent sequences: A394887 A394888 A394889 * A394891 A394892 A394893

Keyword

nonn,tabf

Author

Antti Karttunen, Apr 07 2026

Status

approved