A372880 - OEIS

%I #28 Jun 23 2024 08:38:54

%S 1,3,12,36,612,1836,168912,10810368,16366897152,51703028103168,

%T 1563447866811697152,23520172003575940628103168,

%U 1155558163424267804668132116971520,12369352104691609178206055357839959406281031680

%N a(1) = 1; a(2) = 3; for n > 2, a(n) is the smallest proper multiple of a(n-1) that contains a(n-2) as subsequence of its digits.

%C It is unknown whether a(n+1)/a(n) -> oo as n -> oo.

%C The same rule starting from terms 1, 2 gives A004643 and its multiples are as easy as A004643(n+1)/A004643(n) = 2 or 5 alternately.

%H Kevin Ryde, <a href="/A372880/a372880.c.txt">C Code</a>

%F a(n) <= a(n-2)*10^k + (a(n-1) - (a(n-2)*10^k mod a(n-1))), where k is the number of decimal digits in a(n-1). - _Michael S. Branicky_, May 17 2024

%e a(7) = 168912; 16812 = 92*1836 = 92*a(6) and "16812" contains a(5) = 612 as a subsequence.

%o (Python)

%o def subseq(x,y):

%o     i = 0

%o     j = 0

%o     while i != len(x) and j != len(y):

%o         if x[i] == y[j]:

%o            i += 1

%o         j += 1

%o     return i == len(x)

%o def a(n):

%o     if n == 1:

%o         return 1

%o     A = 1

%o     B = 3

%o     for _ in range(n-2):

%o         s = str(A)

%o         i = 1

%o         while not subseq(s, str(B*i)):

%o             i += 1

%o         A, B = B, B*i

%o     return B

%o (Python)

%o from itertools import count, islice

%o def is_subseq(s, p):

%o     while s and p:

%o         if p%10 == s%10: s //= 10

%o         p //= 10

%o     return s == 0

%o def agen(): # generator of terms

%o     an2, an1 = [1, 3]

%o     yield from [an2, an1]

%o     while True:

%o         an = next(i*an1 for i in count(1) if is_subseq(an2, i*an1))

%o         an2, an1 = an1, an

%o         yield an

%o print(list(islice(agen(), 11))) # _Michael S. Branicky_, May 15 2024

%o (C) /* See links. */

%Y Cf. A004643.

%K nonn,base,more

%O 1,2

%A _Bryle Morga_, May 15 2024

%E a(12)-a(13) from _Michael S. Branicky_, May 15 2024

%E a(14) from _Kevin Ryde_, Jun 23 2024