A392492 a(n) is the smallest p-composable number where p = prime(n).

12, 135, 1645, 14651, 8041, 175253, 115957, 695267, 2128351, 1590331, 5024263, 8294771, 342637, 6319667, 33231397, 44185411, 13384681, 80867639, 92994593, 59796271, 143950087, 215345231, 249369101, 5054221, 175062593, 4646101, 538122367, 423938387, 1075895291, 1109440441

Offset

1,1

Comments

For the definition of a p-composable number see A392440.

Links

Table of n, a(n) for n=1..30.

Peter Luschny, Composable, rigid, and sparse numbers. A Python notebook.

Example

12 is 2-composable as it can be written as the sum of a subset of the divisors of 12 that are > 1 and < 12, namely as 2 + 4 + 6, and {2, 4} is a subset drawn from divisors strictly smaller than 12/2.
.
In the table the minimal p-composable n is given and the lex-earliest witness is shown in form: (lex-earliest witness of n/p) + (p-1)*(n/p).
      n = smallest
  p  |p-composable| n / p  | lexicographically-earliest sum witness
  --------------------------------------------------------------------
  2  |     12     |     6  |  2 +   4 +    6;
  3  |    135     |    45  |  3 +  15 +   27 +    90;
  5  |   1645     |   329  |  5 +   7 +   35 +    47 +   235 +    1316;
  7  |  14651     |  2093  |  7 +  23 +  299 +   637 +  1127 +   12558;
  11 |   8041     |   731  | 11 +  17 +   43 +   187 +   473 +    7310;
  13 | 175253     | 13481  | 17 +  61 +  221 +  2873 + 10309 +  161772;
  17 | 115957     |  6821  | 17 +  19 +  323 +   359 +  6103 +  109136;

Prog

(Python) # Function 'is_composable' is defined in A392440.
# Implement function 'smallest_prime_divisor' A020639, trial division is sufficient.
from sympy import sieve
def first_p_composable(p: int, limit: int = 10**12) -> int | None:
    if p == 2: return 12
    for n in range(p*p, limit + 1, 2 * p):
        if smallest_prime_divisor(n) != p: continue
        if is_composable(n): return n
    return None
def A392492_list(upto: int) -> list[int | None]:
    return [first_p_composable(p) for p in sieve.primerange(2, upto + 1)]
print(A392492_list(113))

Crossrefs

Cf. A392440, A392438 (p=2), A392437 (p=3), A020639.

Sequence in context: A218762 A199233 A085938 * A161124 A288035 A290474

Adjacent sequences: A392489 A392490 A392491 * A392493 A392494 A392495

Keyword

nonn

Author

Peter Luschny, Jan 14 2026

Status

approved