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Let p = first digit of n, q = number obtained if p is removed from n; let r = last digit of n, s = number obtained if r is removed from n; sequence give n such that p*q = r*s != 0, p! = q, and r! = s.
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%I #25 Nov 06 2024 15:26:55

%S 111,164,195,222,265,333,444,498,555,666,777,888,999,1111,1664,1995,

%T 2222,2665,3333,4444,4847,4998,5555,6545,6666,7424,7777,8888,9999,

%U 11111,16664,19995,22222,26665,33333,43243,44444,49998,55555,66666,77777,86486,88888,99999,111111,166664

%N Let p = first digit of n, q = number obtained if p is removed from n; let r = last digit of n, s = number obtained if r is removed from n; sequence give n such that p*q = r*s != 0, p! = q, and r! = s.

%C Once A010785(n) > 100, then A010785 is a subsequence.

%C By the definition, one-digit and two-digit numbers are ruled out.

%H Harvey P. Dale, <a href="/A245364/b245364.txt">Table of n, a(n) for n = 1..74</a> (all terms up to 10 million)

%e 1*64 = 16*4 = 64. Thus 164 is a term of this sequence.

%e 9*999 = 999*9 = 8991. Thus 9999 is a term of this sequence.

%t pqrsQ[n_]:=Module[{p=IntegerDigits[n][[1]],q=FromDigits[Rest[ IntegerDigits[ n]]],r=Mod[n,10],s=Floor[n/10]},p*q==r*s!=0 && p!=q && r!=s]; Select[ Range[100,200000],pqrsQ] (* _Harvey P. Dale_, Aug 29 2020 *)

%o (Python)

%o for n in range(100, 10**5):

%o s = str(n)

%o num = int(s[:1])*int(s[1:])

%o if num != 0 and num == int(s[:len(s)-1])*int(s[len(s)-1:]):

%o print(n, end=', ')

%Y Cf. A010785.

%K nonn,base

%O 1,1

%A _Derek Orr_, Jul 19 2014