Dudeney number

Let ${\displaystyle n}$ be a natural number. We define the Dudeney function for base ${\displaystyle b>1}$ and power ${\displaystyle p>0}$ ${\displaystyle F_{p,b}:\mathbb {N} \rightarrow \mathbb {N} }$ to be the following:

${\displaystyle F_{p,b}(n)=\sum _{i=0}^{k-1}{\frac {n^{p}{\bmod {b^{i+1}}}-n^{p}{\bmod {b}}^{i}}{b^{i}}}}$

where ${\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}$ is the number of digits in the number in base ${\displaystyle b}$.

A natural number ${\displaystyle n}$ is a Dudeney root if it is a fixed point for ${\displaystyle F_{p,b}}$, which occurs if ${\displaystyle F_{p,b}(n)=n}$. The natural number ${\displaystyle m=n^{p}}$ is a generalised Dudeney number,[1] and for ${\displaystyle p=3}$, the numbers are known as Dudeney numbers. ${\displaystyle 0}$ and ${\displaystyle 1}$ are trivial Dudeney numbers for all ${\displaystyle b}$ and ${\displaystyle p}$, all other trivial Dudeney numbers are nontrivial trivial Dudeney numbers.

For ${\displaystyle p=3}$ and ${\displaystyle b=10}$, there are exactly six such integers (sequence A061209 in the OEIS): ${\displaystyle 1,512,4913,5832,17576,19683}$

A natural number ${\displaystyle n}$ is a sociable Dudeney root if it is a periodic point for ${\displaystyle F_{p,b}}$, where ${\displaystyle F_{p,b}^{k}(n)=n}$ for a positive integer ${\displaystyle k}$, and forms a cycle of period ${\displaystyle k}$. A Dudeney root is a sociable Dudeney root with ${\displaystyle k=1}$, and a amicable Dudeney root is a sociable Dudeney root with ${\displaystyle k=2}$. Sociable Dudeney numbers and amicable Dudeney numbers are the powers of their respective roots.

The number of iterations ${\displaystyle i}$ needed for ${\displaystyle F_{p,b}^{i}(n)}$ to reach a fixed point is the Dudeney function's persistence of ${\displaystyle n}$, and undefined if it never reaches a fixed point.

It can be shown that given a number base ${\displaystyle b}$ and power ${\displaystyle p}$, the maximum Dudeney root has to satisfy this bound:

${\displaystyle n\leq (b-1)(1+p+\log _{b}{n^{p}})=(b-1)(1+p+p\log _{b}{n}))}$

implying a finite number of Dudeney roots and Dudeney numbers for each order ${\displaystyle p}$ and base ${\displaystyle b}$.[2]

${\displaystyle F_{1,b}}$ is the digit sum. The only Dudeney numbers are the single-digit numbers in base ${\displaystyle b}$, and there are no periodic points with prime period greater than 1.

All numbers are represented in base ${\displaystyle b}$.

${\displaystyle p}$	${\displaystyle b}$	Nontrivial Dudeney roots ${\displaystyle n}$	Nontrivial Dudeney numbers ${\displaystyle m=n^{p}}$	Cycles of ${\displaystyle F_{p,b}(n)}$	Amicable/Sociable Dudeney numbers
2	2	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	3	2	11	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	4	3	21	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	5	4	31	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	6	5	41	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	7	3, 4, 6	12, 22, 51	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	8	7	61	2 → 4 → 2	4 → 20 → 4
2	9	8	71	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	10	9	81	13 → 16 → 13	169 → 256 → 169
2	11	5, 6, A	23, 33, 91	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	12	B	A1	9 → 13 → 14 → 12	69 → 169 → 194 → 144
2	13	4, 9, C, 13	13, 63, B1, 169	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
2	14	D	C1	9 → 12 → 9	5B → 144 → 5B
2	15	7, 8, E	34, 44, D1	2 → 4 → 2 9 → B → 9	4 → 11 → 4 56 → 81 → 56
2	16	6, A, F	24, 64, E1	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
3	2	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
3	3	11, 22	2101, 200222	12 → 21 → 12	11122 → 110201 → 11122
3	4	2, 12, 13, 21, 22	20, 3120, 11113, 23121, 33220	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
3	5	3, 13, 14, 22, 23	102, 4022, 10404, 23403, 32242	12 → 21 → 12	2333 → 20311 → 2333
3	6	13, 15, 23, 24	3213, 10055, 23343, 30544	11 → 12 → 11	1331 → 2212 → 1331
3	7	2, 4, 11, 12, 14, 15, 21, 22	11, 121, 1331, 2061, 3611, 5016, 12561, 14641	25 → 34 → 25	25666 → 63361 → 25666
3	8	6, 15, 16	330, 4225, 5270	17 → 26 → 17	6457 → 24630 → 6457
3	9	3, 7, 16, 17, 25	30, 421, 4560, 5551, 17618	5 → 14 → 5 12 → 21 → 12 18 → 27 → 18	148 → 3011 → 148 1738 → 6859 → 1738 6658 → 15625 → 6658
3	10	8, 17, 18, 26, 27	512, 4913, 5832, 17576, 19683	19 → 28 → 19	6859 → 21952 → 6859
3	11	5, 9, 13, 15, 18, 22, 25	104, 603, 2075, 3094, 5176, A428, 13874	8 → 11 → 8 A → 19 → A 14 → 23 → 14 16 → 21 → 16	426 → 1331 → 426 82A → 6013 → 82A 2599 → 10815 → 2599 3767 → 12167 → 3767
3	12	19, 1A, 1B, 28, 29, 2A	5439, 61B4, 705B, 16B68, 18969, 1A8B4	8 → 15 → 16 → 11 → 8 13 → 18 → 21 → 14 → 13	368 → 2A15 → 3460 → 1331 → 368 1B53 → 4768 → 9061 → 2454 → 1B53
4	2	11, 101	1010001, 1001110001	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
4	3	11	100111	22 → 101 → 22	12121201 → 111201101 → 12121201
4	4	3, 13, 21, 31	1101, 211201, 1212201, 12332101	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
4	5	4, 14, 22, 23, 31	2011, 202221, 1130421, 1403221, 4044121	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
4	6	24, 32, 42	1223224, 3232424, 13443344	14 → 23 → 14	114144 → 1030213 → 114144
5	2	110, 111, 1001	1111001100000, 100000110100111, 1110011010101001	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
5	3	101	12002011201	22 → 121 → 112 → 110 → 22	1122221122 → 1222021101011 → 1000022202102 → 110122100000 → 1122221122
5	4	2, 22	200, 120122200	21 → 33 → 102 → 30 → 21	32122221 → 2321121033 → 13031110200 → 330300000 → 32122221
6	2	110	1011011001000000	111 → 1001 → 1010 → 111	11100101110010001 → 10000001101111110001 → 11110100001001000000 → 11100101110010001
6	3	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$	101 → 112 → 121 → 101	1212210202001 → 112011112120201 → 1011120101000101 → 1212210202001

Dudeney numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.

The example below implements the Dudeney function described in the definition above to search for Dudeney roots, numbers and cycles in Python.

def dudeneyf(x: int, p: int, b: int) -> int:
    """Dudeney function."""
    y = pow(x, p)
    total = 0
    while y > 0:
        total = total + y % b
        y = y // b
    return total

def dudeneyf_cycle(x: int, p: int, b: int) -> List:
    seen = []
    while x not in seen:
        seen.append(x)
        x = dudeneyf(x, p, b)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = dudeneyf(x, p, b)
    return cycle

• Arithmetic dynamics
• Factorion
• Happy number
• Kaprekar's constant
• Kaprekar number
• Meertens number
• Narcissistic number
• Perfect digit-to-digit invariant
• Perfect digital invariant
• Sum-product number

• H. E. Dudeney, 536 Puzzles & Curious Problems, Souvenir Press, London, 1968, p 36, #120.

• Generalized Dudeney Numbers
• Proving There are Only Six Dudeney Numbers