A392325 Vampire numbers in base 12 (definition 2): decimal values of base-12 integers n for which there exist fangs x and y, where x and y have the same number of base-12 digits, x * y = n, the concatenation of the base-12 digits of x and y is a permutation of the base-12 digits of n, and not both x and y end in 0 (base 12).

2100, 2220, 2352, 2628, 2910, 3052, 3200, 5550, 6264, 8062, 252257, 253071, 253164, 254604, 256188, 257955, 259500, 259522, 262884, 264588, 265005, 267168, 274455, 275237, 275880, 277920, 281060, 281203, 281232, 281418, 281554, 283745, 286006, 286649, 287375

Offset

1,1

References

Clifford A. Pickover, Keys to Infinity, Wiley, NY, 1995, Chap. 30, "Vampire Numbers", pp. 233-238

Links

Lars Verhaevert, Table of n, a(n) for n = 1..10000

Lars Verhaevert, GitHub: Vampire Numbers in base 12 (unsorted outputs, sorted outputs, source codes)

Example

2100 is a term since 1270_12 = 21_12 * 70_12 = 25 *  84 = 2100.
2220 is a term since 1350_12 = 31_12 * 50_12 = 37 *  60 = 2220.
2352 is a term since 1440_12 = 40_12 * 41_12 = 48 *  49 = 2352.
2628 is a term since 1630_12 = 30_12 * 61_12 = 36 *  73 = 2628.
3200 is a term since 1A28_12 = 21_12 * A8_12 = 25 * 128 = 3200.
35883660 is a term since 10026010_12 = 2001_12 * 6010_12 = 3457 * 10380 = 35883660.
35900940 is a term since 10034010_12 = 3001_12 * 4010_12 = 5185 *  6924 = 35900940.
35919948 is a term since 10043010_12 = 3010_12 * 4001_12 = 5196 *  6913 = 35919948.

Prog

(C++) // See GitHub link
(PARI) is(n, b)=my(v=digits(n, b)); if(#v%2, return(0)); fordiv(n, d, if(#digits(d, b)==#v/2 && #digits(n/d, b)==#v/2 && vecsort(v)==vecsort(concat(digits(d, b), digits(n/d, b))) && (d%b || (n/d)%b), return(1))); 0; \\ Michel Marcus, Jun 12 2026
(Python)
from gmpy2 import digits
from sympy import divisors
def ok(n, base=12):
    s = "".join(sorted(digits(n, base)))
    if len(s)&1: return False
    divs, L, U = divisors(n), base**((len(s)//2)-1), base**(len(s)//2)
    for i in range((len(divs)+1)//2+1):
        x, y = divs[i], divs[-1-i]
        if L <= x < U and L <= y < U and (x%base or y%base):
            t, u = digits(x, base), digits(y, base)
            if "".join(sorted(t+u)) == s:
                return True # use return (n, s, t, u) or similar for verbose output
    return False
print([k for k in range(300_000) if ok(k)]) # Michael S. Branicky, Jun 12 2026

Crossrefs

Cf. A014575 (vampire numbers in base 10).

Sequence in context: A045051 A102504 A270761 * A332643 A015293 A159812

Adjacent sequences: A392322 A392323 A392324 * A392326 A392327 A392328

Keyword

nonn,base

Author

Lars Verhaevert, Jun 07 2026

Status

approved