cp's OEIS Frontend

A323142 Envelope numbers (see the Comments section for the definition).

%I A323142 #18 Jan 15 2019 20:30:37
%S A323142 100,101,102,103,104,105,106,107,108,109,200,201,202,203,204,205,206,
%T A323142 207,208,209,300,301,302,303,304,305,306,307,308,309,400,401,402,403,
%U A323142 404,405,406,407,408,409,500,501,502,503,504,505,506,507,508,509,600,601,602,603,604,605,606,607,608,609,700,701,702,703,704
%N A323142 Envelope numbers (see the Comments section for the definition).
%C A323142 An envelope number N has two parts E and C such that C is a multiple of E; E is the concatenation of the first and the last digit of N (the Envelope) and C is the concatenation of the other digits (the Content of the envelope). The integer 12348 is a member of the sequence as 234 (the Content) is a multiple of 18 (the Envelope): indeed 234 = 18*13. Contents that have a leading zero are not admitted (10347 is not a regular envelope number though 34 is a multiple of 17).
%C A323142 Note that some envelope numbers might fit into another envelope (and so on): see the Crossrefs section.
%C A323142 This sequence begins with the same 90 terms of A252480 then differs: A252480(91) = 1000 and A323142(91) = 1100
%H A323142 Jean-Marc Falcoz, <a href="/A323142/b323142.txt">Table of n, a(n) for n = 1..20001</a>
%e A323142 100 has a Content of 0 which is indeed a multiple of the Envelope 10 (0 = 10*0)
%e A323142 101 has a Content of 0 which is indeed a multiple of the Envelope 11 (0 = 11*0)
%e A323142 102 has a Content of 0 which is indeed a multiple of the Envelope 12 (0 = 12*0)
%e A323142 ...
%e A323142 1100 has a Content of 10 which is indeed a multiple of the Envelope 10 (10 = 10*1)
%e A323142 1111 has a Content of 11 which is indeed a multiple of the Envelope 11 (11 = 11*1)
%e A323142 1122 has a Content of 12 which is indeed a multiple of the Envelope 12 (12 = 12*1)
%e A323142 ...
%e A323142 1263 has a Content of 26 which is indeed a multiple of the Envelope 13 (26 = 13*2)
%t A323142 Select[Range[100, 704], Or[#1 == 0, Mod[#1, #2] == 0] & @@ {If[And[First@ # == 0, Length@ # > 1], -1, FromDigits@ #] &@ Most@ Rest@ #, FromDigits@ {First@ #, Last@ #}} &@ IntegerDigits@ # &] (* _Michael De Vlieger_, Jan 07 2019 *)
%o A323142 (PARI) isok(n,base=10) = my (d=digits(n,base)); #d>=3 && (#d==3 || d[2]) && ((n-d[1]*base^(#d-1))\base) % (d[1]*base+d[#d])==0 \\ _Rémy Sigrist_, Jan 06 2019
%Y A323142 Cf. A323143 (envelope numbers that fit into successive bigger envelopes).
%Y A323142 Cf. A252480.
%K A323142 base,nonn
%O A323142 1,1
%A A323142 _Eric Angelini_ and _Jean-Marc Falcoz_, Jan 05 2019