An esthetic number is a positive integer where every adjacent digit differs from its neighbour by 1.



E.G.
 12 is an esthetic number. One and two differ by 1.
 5654 is an esthetic number. Each digit is exactly 1 away from its neighbour.
 890 is not an esthetic number. Nine and zero differ by 9.


These examples are nominally in base 10 but the concept extends easily to numbers in other bases. Traditionally, single digit numbers are included in esthetic numbers; zero may or may not be. For our purposes, for this task, do not include zero (0) as an esthetic number. Do not include numbers with leading zeros.

Esthetic numbers are also sometimes referred to as stepping numbers.



Task
 Write a routine (function, procedure, whatever) to find esthetic numbers in a given base.
 Use that routine to find esthetic numbers in bases 2 through 16 and display, here on this page, the esthectic numbers from index (base × 4) through index (base × 6), inclusive. (E.G. for base 2: 8th through 12th, for base 6: 24th through 36th, etc.)
 Find and display, here on this page, the base 10 esthetic numbers with a magnitude between 1000 and 9999.
 Stretch: Find and display, here on this page, the base 10 esthetic numbers with a magnitude between 1.0e8 and 1.3e8.


Related task

   numbers with equal rises and falls


See also

OEIS A033075 - Positive numbers n such that all pairs of consecutive decimal digits differ by 1
Numbers Aplenty - Esthetic numbers
Geeks for Geeks - Stepping numbers


#include <stdio.h> 
#include <string.h>
#include <locale.h>
 
typedef int bool;
typedef unsigned long long ull;
 
#define TRUE 1
#define FALSE 0
 
char as_digit(int d) { 
    return (d >= 0 && d <= 9) ? d + '0' : d - 10 + 'a';  
}
 
void revstr(char *str) { 
    int i, len = strlen(str);
    char t; 
    for (i = 0; i < len/2; ++i) { 
        t = str[i]; 
        str[i] = str[len - i - 1]; 
        str[len - i - 1] = t; 
    } 
}  
 
char* to_base(char s[], ull n, int b) { 
    int i = 0; 
    while (n) { 
        s[i++] = as_digit(n % b); 
        n /= b; 
    } 
    s[i] = '\0'; 
    revstr(s);
    return s;  
} 
 
ull uabs(ull a, ull  b) {
    return a > b ? a - b : b - a;
}
 
bool is_esthetic(ull n, int b) {
    int i, j;
    if (!n) return FALSE;
    i = n % b;
    n /= b;
    while (n) {
        j = n % b;
        if (uabs(i, j) != 1) return FALSE;
        n /= b;
        i = j;
    }
    return TRUE;
}
 
ull esths[45000];
int le = 0;
 
void dfs(ull n, ull m, ull i) {
    ull d, i1, i2;
    if (i >= n && i <= m) esths[le++] = i;
    if (i == 0 || i > m) return; 
    d = i % 10;
    i1 = i * 10 + d - 1;
    i2 = i1 + 2;
    if (d == 0) {
        dfs(n, m, i2);
    } else if (d == 9) {
        dfs(n, m, i1);
    } else {
        dfs(n, m, i1);
        dfs(n, m, i2);
    }
}
 
void list_esths(ull n, ull n2, ull m, ull m2, int per_line, bool all) {
    int i;
    le = 0;
    for (i = 0; i < 10; ++i) {
        dfs(n2, m2, i);
    }
    printf("Base 10: %'d esthetic numbers between %'llu and %'llu:\n", le, n, m);
    if (all) {
        for (i = 0; i < le; ++i) {
            printf("%llu ", esths[i]);
            if (!(i+1)%per_line) printf("\n");
        }
    } else {
        for (i = 0; i < per_line; ++i) printf("%llu ", esths[i]);
        printf("\n............\n");
        for (i = le - per_line; i < le; ++i) printf("%llu ", esths[i]);
    }
    printf("\n\n");
}
 
int main() {
    ull n;
    int b, c;
    char ch[15] = {0};
    for (b = 2; b <= 16; ++b) {
        printf("Base %d: %dth to %dth esthetic numbers:\n", b, 4*b, 6*b);
        for (n = 1, c = 0; c < 6 * b; ++n) {
            if (is_esthetic(n, b)) {
                if (++c >= 4 * b) printf("%s ", to_base(ch, n, b));
            }
        }
        printf("\n\n");
    }
    char *oldLocale = setlocale(LC_NUMERIC, NULL);
    setlocale(LC_NUMERIC, ""); 
 
    // the following all use the obvious range limitations for the numbers in question
    list_esths(1000, 1010, 9999, 9898, 16, TRUE);
    list_esths(1e8, 101010101, 13*1e7, 123456789, 9, TRUE);
    list_esths(1e11, 101010101010, 13*1e10, 123456789898, 7, FALSE);
    list_esths(1e14, 101010101010101, 13*1e13, 123456789898989, 5, FALSE);
    list_esths(1e17, 101010101010101010, 13*1e16, 123456789898989898, 4, FALSE);
    setlocale(LC_NUMERIC, oldLocale);
    return 0;
}