A316680
The integer 1358 and its infinite continuation (when iterating the rule explained in A316650 and in the Comment section here).
Original entry on oeis.org
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
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.
Cf.
A316650 (where the rule is explained).
Cf.
A316679 (for an equivalent pattern produced by 907).
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NestList[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, 1358, 28] (* Michael De Vlieger, Jul 10 2018 *)
A316678
Smallest numbers leading in n steps to a term that repeats itself, according to the rule explained in A316650 (and hereunder in the Comment section).
Original entry on oeis.org
19, 31, 13, 16, 32, 11, 23, 15, 236, 282, 341, 1047, 787, 419, 286, 626, 557, 498, 1357, 1001, 368, 1921, 917, 2077, 3319, 3457, 5090, 2294, 2144, 3501, 4485, 10661, 16753, 3092, 5252, 3475, 2102, 3572, 656, 1691, 7461, 10445, 4596, 13937, 15964, 25540, 14380
Offset: 1
19 is the smallest integer leading to itself in 1 step because we have [19/10 = 10*1 + 9];
31 is the smallest integer ending on a fixed point in 2 steps because 31 leads to 73 [31/4 = 4*7 + 3] (step 1) and 73 to itself [73/10 = 10*7 + 3] (step 2);
13 is the smallest integer ending on a fixed point in 3 steps because 13 leads to 31 [13/4 = 4*3 + 1] (step 1) and 31 leads to 73 in 2 steps (see above);
16 is the smallest integer ending on a fixed point in 4 steps because 16 leads to 22 [16/7 = 7*2 + 2] (step 1), then 22 leads to 52 [22/4 = 4*5+2] (step 2), then 52 leads to 73 [52/7 = 7*7 + 3] (step 3) and 73 to itself [73/10 = 10*7 + 3] (step 4);
32 is the smallest integer ending on a fixed point in 5 steps [32,62,76,511,730];
11 is the smallest integer ending on a fixed point in 6 steps
[11,51,83,76,511,730];
23 is the smallest integer ending on a fixed point in 7 steps
[23,43,61,85,67,52,73];
Etc.
Cf.
A316650 (where the main idea is explained),
A316679 (for the infinite growing pattern produced by 907) and
A316680 (for the infinite growing pattern produced by 1358).
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Array[Block[{k = 1}, While[Count[#, 0] != 1 &@ Differences@ NestList[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, k, #], k++]; k] &, 47] (* Michael De Vlieger, Jul 10 2018 *)
Showing 1-2 of 2 results.
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