A381732 Proceeding from left to right, between any two consecutive digits (d_i, d_i+1) of an integer k, write down apart the lacking consecutive digits, in increasing order if d_i d_i+1. If abs(d_i - d_i+1) = 0 or 1 no digit is added. Sequence lists integers k that divide such resulting numbers.

27, 737, 909, 1845, 1912, 7078, 27412, 90009, 870129, 990099, 6852899, 9090909, 17388261, 70168376, 70787078, 96096078, 96707298, 162533711, 358006673, 737737737, 1050889491, 2238028254, 3281718034, 4249370147, 9009009009, 11819327599, 12178217823, 13851266943, 18768863945

Offset: 1

Author

Paolo P. Lava, Mar 05 2025

Keywords

• nonn
• base

Comments

These concatenations are part of the sequence:

'737' with itself, if it is not a multiple of 7;

'7078' with itself, if it is not a multiple of 3.

Examples

			27 is a term since between 2 and 7 we have 3456 and 3456 / 27 = 128;
1845 is a term since between 1 and 8 we have 234567, between 8 and 4 765 and between 4 and 5 no digit to be added and 234567765 / 1845 = 127137.
		

Links

• Carlos Rivera, Puzzle 1212 A381732, The Prime Puzzles and Problems Connection.

Programs

Python

def f(n):
    s, out = list(map(int, str(n))), 0
    for i in range(len(s)-1):
        dir = 1 if s[i+1] - s[i] >= 0 else -1
        for j in range(s[i]+dir, s[i+1], dir):
            out = 10*out + j
    return out
def ok(n):
    return (v:=f(n)) and v%n == 0
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Mar 06 2025

Extensions

a(19)-a(29) from Michael S. Branicky, Mar 07 2025