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

2,1

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.

Links

Serguei Zolotov, Antidiagonals of table of n, a(n) for n = 2..2081

Serguei Zolotov, Python script to generate A341339 combining different methods

Example

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, ...
...

Prog

(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

Crossrefs

Sequence in context: A305622 A269940 A350463 * A084257 A365803 A185963

Adjacent sequences: A341336 A341337 A341338 * A341340 A341341 A341342

Keyword

nonn,tabl

Author

Serguei Zolotov, Apr 27 2021

Status

approved