A384537 Composite numbers that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

16, 27, 64, 256, 729, 1024, 3125, 4096, 4617, 16384, 19683, 29767, 65536, 255987, 262144, 395847, 531441, 631463, 823543, 1048576, 1332331, 4194304, 9765625, 14348907, 16777216, 25640947, 67108864

Offset: 1

Author

Jianing Song, Jun 02 2025

Keywords

• nonn
• hard
• more

Comments

Someone called James Davis found that 13532385396179 = 13 * 53^2 * 3853 * 96179, showing that a composite number can be equal the concatenation of the primes and exponents in its canonical prime factorization. In general, if a composite number is equal the concatenation in base b of the primes and exponents in its prime factorization, then let's call it a Davis number to base b.

Conjecture: a composite number can be a Davis number to at most one base.

Let (d_1,...,d_r) be the ordered tuple of prime factors and exponents > 1 in the prime factorization of n (e.g., 4617 = 3^5 * 19 -> (3,5,19), 13532385396179 = 13 * 53^2 * 3853 * 96179 -> (13,53,2,3853,96179)), then n is a Davis number to base b if and only if n = d_1*b^{s_1} + ... + d_{r-1}*b^{s_{r-1}} + d_r, where s_i = (Sum_{j=i+1..r} floor(log_b(d_j))) + r-i. In particular, we must have b dividing n - d_r.

Suppose that p^e is a Davis number to some base b, with e >= 2. We have p^e = p*b^(floor(log_b(e))+1) + e in base b, hence e is divisible by p. If b <= e, then we have p^e <= p*b^(log_b(e)+1) + e <= p*e^2 + e, which is impossible, and so we must have b > e. Conversely, when e is divisible by p and p^e > 4, p^e is a Davis number to base (p^e-e)/p > e.

No term can be squarefree: for primes p_1 < ... < p_r, the concatenation of p_1, ..., p_r in base b is p_1 * b^(Sum_{i=2..r} (floor(log_b(p_i))+1)) + ... >= p_1*...*p_r + ... > p_1*...*p_r.

Here are some examples that are near-miss of being Davis numbers to base 10. Each is equal to the concatenation of the factors and exponents in its generalized factorization (we call n = (q_1)^(e_1) * ... * (q_k)^(e_k) a generalized factorization of n, where 1 < q_1 < ... < q_k, (q_1,...,q_k) are pairwise coprime but are not necessarily primes, and exponents 1 are omitted; the number of such factorizations is A327399(n)):

2592 = 2^5 * 9^2;

34425 = 3^4 * 425;

312325 = 31^2 * 325;

492205 = 49^2 * 205;

36233196159122085048010973936921313644799483579440006455257 = 3^6 * 2331961591220850480109739369 * 21313644799483579440006455257. (Note that in the last four examples, we can add as many trailing zeros as we want).

Examples

			In base 6: 24 = 2^4 (in decimal: 16 = 2^4);
In base 8: 33 = 3^3 (in decimal: 27 = 3^3);
In base 29: 26 = 2^6 (in decimal: 64 = 2^6);
In base 124: 28 = 2^8 (in decimal: 256 = 2^8);
In base 241: 36 = 3^6 (in decimal: 729 = 3^6);
In base 507: 2A = 2^A (in decimal: 1024 = 2^10);
In base 624: 55 = 5^5 (in decimal: 3125 = 5^5);
In base 2042: 2C = 2^C (in decimal: 4096 = 2^12);
In base 11: 3518 = 3^5 * 18 (in decimal: 4617 = 3^5 * 19).
See A384540 for more nontrivial examples.
		

Crossrefs

Cf. A080670, A195264, A230625, A384540, A327399.

Programs

PARI

F(n,b) = my(f=factor(n), d=[]); for(i=1, #f~, d=concat(d, digits(f[i,1],b)); if(f[i,2]>1, d=concat(d, digits(f[i,2],b)))); fromdigits(d,b)
isA384537(n) = {
if(issquarefree(n), return(0)); my(f=factor(n), dr);
if(#f~ == 1, return(n > 4 && f[1,2] % f[1,1] == 0));
dr = if(f[#f~,2] == 1, f[#f~,1], f[#f~,2]);
fordiv(n - dr, b, if(b>=2 && F(n,b)==n, return(b))); return(0);
} \\ returns the (smallest) base to which n is a Davis number whenever possible