A248323 - OEIS

%I #27 Jan 02 2023 12:30:50

%S 11,12,42,84,124,135,248,325,366,510,550,555,624,650,714,1010,1111,

%T 1525,1734,2510,3913,4020,5100,5500,5610,5625,8040,11111,13515,16575,

%U 21175,24104,25500,28160,34170,35250,35610,36800,37444,44919,50100,51020,51102,51250,52000

%N Numbers n which appear at least twice in A037278(n), concatenation of their divisors written in base 10.

%C Overlapping is allowed, so a(1) = 11 is in the sequence, with concatenated divisors A037278(11) = "111".

%C All repunits (10^k-1)/9 = A000042(k) = A002275(k) with even k = number of digits (as to be divisible by 11) but not multiples of 3, i.e., k in A047235, have divisors {1, 11, ..., 1010...101, 1111...111} and therefore are in this sequence.

%C Numbers n = floor(10^(8+3k)/7), k>=0, also belong to this sequence; for k>=2m, the number n appears (at least) m+2 times in A037278(n). [Found by extending results from _Hans Havermann_.]

%C The smallest terms that appear more than twice in the concatenation are 1111, 400200, 800400, 28571428, all 3 times, and 42857142, 4 times. - _Hans Havermann_, Oct 05 2014

%H Chai Wah Wu, <a href="/A248323/b248323.txt">Table of n, a(n) for n = 1..340</a>

%H E. Angelini, <a href="http://list.seqfan.eu/oldermail/seqfan/2014-October/013737.html">Divisors showing a string</a>, SeqFan list, Oct 03 2014

%e The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42, and "42" appears twice in their concatenation A037278(42) = "12367142142".

%o (PARI) is(n)={d=concat(apply(digits,divisors(n)));n=digits(n);for(j=0,#d-#n-1,for(i=1,#n,d[i+j]==n[i]||next(2));return(1))}

%o (Python)

%o from sympy import divisors

%o import re

%o A248323_list = [n for n in range(1,10**7) if len(list(re.finditer('(?='+str(n)+')',''.join([str(d) for d in divisors(n)])))) > 1]

%o # _Chai Wah Wu_, Nov 01 2014

%Y Cf. A037278.

%K nonn,base

%O 1,1

%A _Eric Angelini_ and _M. F. Hasler_, Oct 04 2014