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A322322
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Stuffable numbers (see the Comments section for definition).
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4
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10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 29, 30, 33, 34, 35, 37, 38, 40, 42, 43, 44, 45, 48, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 72, 74, 75, 77, 80, 81, 82, 83, 87, 88, 89, 90, 91, 92, 94, 96, 98, 99, 100, 101
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OFFSET
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1,1
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COMMENTS
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For given n > 9, consider integers k whose number of digits equals the sum of digits of n minus its last digit. If (d[i])_{i=1..N} are the digits of n, let S(n,k) be the result of inserting d[i] digits of k after the i-th digit of n, i = 1, ..., N-1. If S(n,k) is divisible by k, then n is called a stuffable number. Let s(n) be the set of all such k, for given n. If s(n) has m elements, then n is called m-stuffable. If n is in s(n), n is called self-stuffable. See A322323 for the list of all self-stuffable numbers.
More generally, any n = 10^r + s with 0 <= s <= 9 is in the sequence, since it can be "stuffed" with k = 1, i.e., S(n,1) = 11*10^r + s is divisible by 1.
The following is a heuristic for finding terms. Let f be a value such that f * "stuffable number" = "stuffed number." For the example 2018, f = 15663. If a number is stuffable, we can find lower and upper bounds on f. For each f in that interval we test if some stuff could produce that f.
We then start with stuff of the form 10^k and update the "stuff" and "stuffable" until we know f can't produce a stuffable.
For 2018 we know floor(2990198 / 999) = 2993 <= f <= 21002 = ceiling(2100108/100).
The iterations with stuff and stuffed respectively look as follows, for f = 3108:
100 2100108
676 2670168
860 2860108
921 2920118
940 2940108
2018: 2940168 = 946 * 3108
The heuristic as is needs modifications to become an algorithm. (End)
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LINKS
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EXAMPLE
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We explain what a "stuffable number" is with an example. Take 2018 and "open" it like this: 2018 becomes 2..01.8 (two spaces after 2, no space after the 0, one space after 1, nothing after 8 -- and, in general, k spaces after each digit k, except for the last one).
One then "pushes" an integer into the empty spaces; here, for instance, we push 136 and 2..01.8 becomes 2130168.
Let's call 136 the stuff, and 2130168 the stuffed number. If the stuffed number is divisible by the stuff, we will call the starting integer a "stuffable number".
This is the case for 2018, as 2130168 divided by 136 gives 15663.
Remark: no stuff can begin with a leading zero.
Simpler examples are given by the first terms of this sequence:
10 is in the sequence as 10 will be "opened" as 1.0 and the dot can be filled by the integer 1 (as 1 divides 110).
11 through 19 are in the sequence for the same reason. (See also the second comment.)
Some stuffable numbers admit more than one stuff: for instance, 20 becomes 2..0 and the 6 following stuffed numbers are divisible by their respective stuffs:
2100 divided by 10 is 210
2160 divided by 16 is 135
2200 divided by 20 is 110
2250 divided by 25 is 90
2400 divided by 40 is 60
2500 divided by 50 is 50
2800 divided by 80 is 35
So, 20 can be filled with any of the seven 2-digit numbers {10, 16, 20, 25, 40, 50, 80}. We could say that 20 is a 7-stuffable number, as 7 stuffs are possible for 20.
21 is in the sequence as 21 will be opened as 2..1 and the dots can be filled by the integer 23, as 23 divides 2231 with the result 97.
22 is in the sequence as 22 will be opened as 2..2 and the dotes can be filled by the integer 11, as 11 divides 2112 with the result 192.
23 is not in the sequence as 23 will be opened as 2..3 and no integer k from 10 to 99 will divide 2k3;
etc.
380 through 384 are terms:
380: 39678775000000 = 96777500000 * 410
381: 34348238156211 = 43423815621 * 791
382: 35098410898382 = 50941089838 * 689
383: 31688466060933 = 16846606093 * 1881
384: 32618788025894 = 26178802589 * 1246 (End)
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MATHEMATICA
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Position[Array[Block[{w = #, s = Position[#, {}][[All, 1]]}, SelectFirst[Range[10^(Length@ s - 1), 10^Length@ s - 1], Function[k, Mod[FromDigits@ ReplacePart[w, Array[s[[#]] -> k[[#]] &, Length@ s]], #] == 0]@ IntegerDigits[#] &] ] &@ Drop[Flatten[Map[Prepend[ConstantArray[{}, #], #] &, #], 1], -Last@ #] &@ IntegerDigits[#] &, 70], _?IntegerQ][[All, 1]] (* Michael De Vlieger, Dec 08 2018 *)
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PROG
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(PARI) is(n, d=digits(n), D=vecsum(d)-d[#d], i)=for(k=10^(D-1), 10^D-1, D=digits(k); i=0; fromdigits(concat(vector(#d, j, if( d[j]&& j<#d, vector(d[j]+1, k, if(k>1, D[i+=1], d[j])), d[j]))))%k||return(k)) \\ M. F. Hasler, Jan 01 2019
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CROSSREFS
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The sequence A322323 gives the self-stuffable numbers.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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