A387904 Numbers in which there exist two, distinct, non-overlapping prime substrings that can be swapped to form a prime number.

32, 35, 37, 73, 112, 113, 133, 137, 172, 173, 175, 197, 232, 235, 251, 253, 272, 275, 277, 292, 305, 311, 313, 317, 319, 322, 323, 325, 329, 331, 332, 335, 337, 343, 347, 359, 361, 362, 365, 367, 373, 377, 379, 382, 392, 395, 412, 415, 437, 473, 475, 521, 527, 532

Offset

1,1

Comments

Leading zeros are disallowed in both substrings chosen for swapping.

No term is divisible by 3. - David A. Corneth, Oct 30 2025

Terms cannot end in 0, 4, 6 or 8; terms ending in 2 or 5 must swap 2 or 5, respectively. - Michael S. Branicky, Oct 30 2025

Links

Table of n, a(n) for n=1..54.

James S. DeArmon, R code

Example

The first term is 32: swapping the digits yields 23.
The first 3-digit term is 112: swapping 11 and 2 yields 211, a prime.
382 is a term as swapping distinct substrings 2 and 3 yields 283 which is prime. - David A. Corneth, Oct 30 2025

Prog

(Python)
from sympy import isprime
def ok(n):
    s = str(n)
    return any(isprime(int(s[:i]+s[j:j+m]+s[i+h:j]+s[i:i+h]+s[j+m:]))
        for i in range(len(s)-1)
        if s[i] != '0'
        for j in range(i+1, len(s))
        if s[j] != '0'
        for h in range(1, j-i+1)
        if isprime(int(s[i:i+h]))
        for m in range(1, len(s)-j+1)
        if s[j:j+m] != s[i:i+h] and isprime(int(s[j:j+m])))
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, Oct 28 2025

Crossrefs

Cf. A000040.

Sequence in context: A074287 A223589 A159889 * A345490 A095481 A095475

Adjacent sequences: A387901 A387902 A387903 * A387905 A387906 A387907

Keyword

nonn,base

Author

James S. DeArmon, Oct 10 2025

Status

approved