A392439 Prime numbers p whose predecessor k = (p - 1) is a composable number (A392440).

13, 19, 31, 37, 41, 43, 61, 67, 73, 79, 97, 101, 103, 109, 113, 127, 139, 151, 157, 163, 181, 193, 197, 199, 211, 223, 229, 241, 271, 277, 281, 283, 307, 313, 331, 337, 349, 353, 367, 373, 379, 397, 401, 409, 421, 433, 439, 449, 457, 461, 463, 487, 491, 499, 521

Offset

1,1

Comments

Numbers k are defined as composable or p-composable if k/p is a sum of nontrivial divisors, where p is the smallest prime divisor of k. Since any p-composable number with p greater than 2 is odd, and the predecessor of any odd prime is even, only 2-composable numbers can precede primes.

A prime p, where k = p - 1 is 2-composable, can be expressed as p = 1 + Sum(S) + k/2. Here, S is a subset of k's divisors strictly between 1 and k/2 that sums to k/2. Including k/2 in S produces a subset of divisors of k less than k that sums to k, thereby proving that k is semiperfect.

Among the 9592 primes less than 10^5, 6043 are preceded by a composable number. Therefore, approximately 63% of the primes in this range possess such a sum representation.

A prime p is a term if it can be written as p = j*m+1 where m is a semiperfect number (A005835) and j > 1. The converse is not true: 491, for example. - Peter Munn, Mar 20 2026

Links

Table of n, a(n) for n=1..55.

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

Example

13 is a composable-preceded prime. Its predecessor is 12 and since the nontrivial divisors 2 and 4 of 12 sum to 12/2, we can write the prime as 13 = 1 + (2 + 4) + 6.
.
  p    | Witness of composable-preceded prime
  -------------------------------------------
  13   | 1 +  2 +  4 +  6
  19   | 1 +  3 +  6 +  9
  31   | 1 +  5 + 10 + 15
  37   | 1 +  6 + 12 + 18
  41   | 1 +  2 +  8 + 10 + 20
  43   | 1 +  7 + 14 + 21
  61   | 1 + 10 + 20 + 30
  67   | 1 + 11 + 22 + 33
  73   | 1 + 12 + 24 + 36
  79   | 1 + 13 + 26 + 39
  97   | 1 + 16 + 32 + 48
  101  | 1 +  5 + 20 + 25 + 50

Prog

(Python)  # Function 'is_composable' is defined in A392440 (see also links).
from sympy import sieve
def A392439_list(upto: int) -> list[int]:
    primes = []
    for p in sieve.primerange(2, upto + 1):
        if is_composable(p - 1):
            primes.append(p)
    return primes
print(A392439_list(522))

Crossrefs

Cf. A000040, A392440 (composable), A392438 (2-composable), A005835 (semiperfect), A032741 (proper divisors), A070824 (nontrivial divisors).

Sequence in context: A108097 A102764 A164318 * A307533 A092738 A272382

Adjacent sequences: A392436 A392437 A392438 * A392440 A392441 A392442

Keyword

nonn

Author

Peter Luschny, Jan 12 2026

Status

approved