A341339 Square array read by descending antidiagonals where the row n (n >= 2) and column k (k >= 1) contains the largest number not greater than 2^k that has exactly n divisors, or 0 if such a number does not exist.

2, 3, 0, 7, 4, 0, 13, 4, 0, 0, 31, 9, 8, 0, 0, 61, 25, 15, 0, 0, 0, 127, 49, 27, 16, 0, 0, 0, 251, 121, 62, 16, 12, 0, 0, 0, 509, 169, 125, 16, 32, 0, 0, 0, 0, 1021, 361, 254, 81, 63, 0, 0, 0, 0, 0, 2039, 961, 511, 81, 124, 64, 30, 0, 0, 0, 0, 4093, 1849, 1018, 81, 245, 64, 56, 0, 0, 0, 0, 0

Offset: 2

Author

Serguei Zolotov, Apr 27 2021

Keywords

• nonn
• tabl

Comments

First row contains largest prime not greater than 2^k (where k is a column number starting with 1). Second row contains largest square of prime not greater than 2^k.

Diagonal of the square array contains sequential powers of 2 since 2^k has exactly k+1 divisors.

Examples

			Array begins:
     k = 1   2   3    4    5    6    7    8    9    10    11    12
-------------------------------------------------------------
n = 2  | 2,  3,  7,  13,  31,  61, 127, 251, 509, 1021, 2039, 4093, ...
n = 3  | 0,  4,  4,   9,  25,  49, 121, 169, 361,  961, 1849, 3721, ...
n = 4  | 0,  0,  8,  15,  27,  62, 125, 254, 511, 1018, 2047, 4087, ...
n = 5  | 0,  0,  0,  16,  16,  16,  81,  81,  81,  625,  625, 2401, ...
n = 6  | 0,  0,  0,  12,  32,  63, 124, 245, 508, 1017, 2043, 4084, ...
n = 7  | 0,  0,  0,   0,   0,  64,  64,  64,  64,  729,  729,  729, ...
n = 8  | 0,  0,  0,   0,  30,  56, 128, 255, 506, 1023, 2037, 4094, ...
n = 9  | 0,  0,  0,   0,   0,  36, 100, 256, 484,  676, 1521, 3844, ...
n = 10 | 0,  0,  0,   0,   0,  48, 112, 208, 512,  976, 2032, 4016, ...
n = 11 | 0,  0,  0,   0,   0,   0,   0,   0,   0, 1024, 1024, 1024, ...
n = 12 | 0,  0,  0,   0,   0,  60, 126, 234, 500, 1014, 2048, 4086, ...
n = 13 | 0,  0,  0,   0,   0,   0,   0,   0,   0,    0,    0, 4096, ...
...
		

Links

• Serguei Zolotov, Antidiagonals of table of n, a(n) for n = 2..2081
• Serguei Zolotov, Python script to generate A341339 combining different methods

Programs

Python

import sympy
# k = 1,2,3,...
# n = 2,3,4,...
def a(k, n):
    a = 2**k
    while a > 0 and sympy.divisor_count(a) != n:
        a = a - 1
    return a