A316680 The integer 1358 and its infinite continuation (when iterating the rule explained in A316650 and in the Comment section here).

1358, 7915, 35917, 143617, 65281, 29677, 95710, 435010, 334624, 152104, 117004, 90004, 69235, 276910, 1107610, 6922510, 27690010, 110760010, 692250010, 2769000010, 11076000010, 69225000010, 276900000010, 1107600000010, 6922500000010, 27690000000010, 110760000000010, 692250000000010, 2769000000000010

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Jul 10 2018

Keywords

• base
• nonn

Comments

It is conjectured, when iterating the idea explained in A316650 ("Result when n is divided by the sum of its digits and the resulting integer is concatenated with the remainder"), that all integers will end either on a fixed point (the first ones are listed in A052224) or grow forever (like 907 or 1358).

Examples

			1358/17 gives 79 with remainder 15;
7915/22 gives 359 with remainder 17;
35917/25 gives 1436 with remainder 17;
143617/22 gives 6528 remainder 1;
...
After 6922510 starts a devilish inflation "from the middle", in a ternary cycle:
6922510
27690010
110760010
692250010
2769000010
11076000010
69225000010
276900000010
1107600000010
6922500000010
27690000000010
110760000000010
692250000000010
2769000000000010
11076000000000010
69225000000000010
276900000000000010
1107600000000000010
6922500000000000010
...
We have:
2769(k zeros)10
11076(k zeros)10
69225(k zeros)10
then:
2769(k+2 zeros)10
11076(k+2 zeros)10
69225(k+2 zeros)10
then:
2769(k+4 zeros)10
11076(k+4 zeros)10
69225(k+4 zeros)10
Etc.
		

Crossrefs

Cf. A316650 (where the rule is explained).

Cf. A316679 (for an equivalent pattern produced by 907).

Programs

Mathematica

NestList[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, 1358, 28] (* Michael De Vlieger, Jul 10 2018 *)