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

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 400, 401, 402, 403, 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

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Jan 05 2019

Keywords

• base
• nonn

Comments

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).

Note that some envelope numbers might fit into another envelope (and so on): see the Crossrefs section.

This sequence begins with the same 90 terms of A252480 then differs: A252480(91) = 1000 and A323142(91) = 1100

Examples

			100 has a Content of 0 which is indeed a multiple of the Envelope 10 (0 = 10*0)
101 has a Content of 0 which is indeed a multiple of the Envelope 11 (0 = 11*0)
102 has a Content of 0 which is indeed a multiple of the Envelope 12 (0 = 12*0)
...
1100 has a Content of 10 which is indeed a multiple of the Envelope 10 (10 = 10*1)
1111 has a Content of 11 which is indeed a multiple of the Envelope 11 (11 = 11*1)
1122 has a Content of 12 which is indeed a multiple of the Envelope 12 (12 = 12*1)
...
1263 has a Content of 26 which is indeed a multiple of the Envelope 13 (26 = 13*2)
		

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..20001

Crossrefs

Cf. A323143 (envelope numbers that fit into successive bigger envelopes).

Cf. A252480.

Programs

Mathematica

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 *)

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