A230625 -id:A230625 - cp's OEIS frontend

A289667 Concatenate prime factorization written in base 3, convert back to decimal.

1, 2, 3, 8, 5, 21, 7, 21, 11, 23, 11, 75, 13, 25, 32, 22, 17, 65, 19, 77, 34, 65, 23, 192, 17, 67, 30, 79, 29, 194, 31, 23, 92, 71, 52, 227, 37, 73, 94, 194, 41, 196, 43, 227, 104, 77, 47, 201, 23, 71, 98, 229, 53, 192, 146, 196, 100, 191, 59, 680, 61, 193, 106, 24

Offset: 1

N. J. A. Sloane, Jul 27 2017

A080670 is the base 10 version, A230625 is the binary version.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A080670, A195264, A230625, A230627.

# take ifsSorted from A080670
A289667 := proc(n)
    local Ldgs, p,eb,pb,b ;
    b := 3;
    if n = 1 then
        return 1;
    end if;
    Ldgs := [] ;
    for p in ifsSorted(n) do
        pb := convert(op(1,p),base,b) ;
        Ldgs := [op(pb),op(Ldgs)] ;
        if op(2, p) > 1 then
            eb := convert(op(2,p),base,b) ;
            Ldgs := [op(eb),op(Ldgs)] ;
        end if;
    end do:
    add( op(e,Ldgs)*b^(e-1),e=1..nops(Ldgs)) ;
end proc:
seq(A289667(n),n=1..30) ; # R. J. Mathar, Aug 05 2017

Table[FromDigits[#, 3] &@ Flatten@ Map[IntegerDigits[#, 3] &, FactorInteger[n] /. {p_, e_} /; p > 0 :> If[e == 1, p, {p, e}]], {n, 64}] (* Michael De Vlieger, Jul 29 2017 *)

a(n) = {if (n==1, return(1)); f = factor(n); s = []; for (i=1, #f~, s = concat(s, digits(f[i, 1], 3)); if (f[i, 2] != 1, s = concat(s, digits(f[i, 2], 3))););  fromdigits(s, 3);} \\ Michel Marcus, Jul 27 2017

More terms from Michel Marcus, Jul 27 2017

A245270 Like A067599 but write everything in binary, then display the answer in base 10.

Original entry on oeis.org

5, 7, 10, 11, 47, 15, 11, 14, 91, 23, 87, 27, 95, 123, 20, 35, 94, 39, 171, 127, 183, 47, 95, 22, 187, 15, 175, 59, 763, 63, 21, 247, 355, 191, 174, 75, 359, 251, 187, 83, 767, 87, 343, 235, 367, 95, 167, 30, 182, 483, 347, 107, 95, 375, 191, 487, 379, 119

Offset: 2

Chai Wah Wu, Jul 15 2014

The only fixed point < 10^8 is 470367 = 3^4 * 5807^1. - Christopher Scussel, Apr 28 2025

			24 = 2^3 * 3^1 has binary encoding 10_11_11_1, that is, 95 in decimal.

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Cf. A067599, A230625.

a(n) = {f = factor(n); s = []; for (i=1, #f~, s = concat(s, binary(f[i, 1])); s = concat(s, binary(f[i, 2]));); subst(Pol(s), x, 2);} \\ Michel Marcus, Jul 16 2014

import sympy
[int(''.join([bin(y)[2:] for x in sorted(sympy.ntheory.factorint(n).items()) for y in x]),2) for n in range(2,200)] # compute a(n) for n > 1
# Chai Wah Wu, Jul 15 2014

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

Original entry on oeis.org

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

Jianing Song, Jun 02 2025

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).

			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.

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

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

A384540 Numbers in A384537 that are not prime powers: composite numbers, not being prime powers, that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

Original entry on oeis.org

4617, 29767, 255987, 395847, 631463, 1332331, 25640947

Offset: 1

Jianing Song, Jun 02 2025

See A384537 for more information.

			In base 11: 3518 = 3^5 * 18 (in decimal: 4617 = 3^5 * 19);
In base 12: 15287 = 15^2 * 87 (in decimal: 29767 = 17^2 * 103);
In base 2: 111110011111110011 = 11^11 * 10011 * 111110011 (in decimal: 255987 = 3^3 * 19 * 499);
In base 362: (3,7,181)_362 = 3^7 * 181.
In base 300: (7,4,263)_300 = 7^4 * 263.
In base 57: 7B4D = 7 * B^4 * D (in decimal: 1332331 = 7 * 11^4 * 13).
In base 1228: (17,4,307)_1228 = 17^4*307.

Cf. A080670, A195264, A230625, A384537.

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)
isA384540(n) = {
if(issquarefree(n), return(0)); my(f=factor(n), dr);
if(#f~ == 1, return(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

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A289667 Concatenate prime factorization written in base 3, convert back to decimal.

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A245270 Like A067599 but write everything in binary, then display the answer in base 10.

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

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PARI

A384540 Numbers in A384537 that are not prime powers: composite numbers, not being prime powers, that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

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