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A317093
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Terms resulting from application of a divisor sieve to the digits of the decimal expansions of the positive integers.
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0
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0, 3, 4, 6, 7, 8, 9, 0, 2, 2, 6, 8, 0, 3, 3, 3, 9, 0, 4, 4, 0, 5, 0, 6, 6, 6, 6, 0, 7, 7, 0, 8, 8, 8, 0, 9, 9, 9, 9, 9, 0, 0, 0, 3, 0, 0, 6, 7, 8, 9, 0, 3, 4, 6, 7, 8, 9, 0, 2, 2, 2, 27, 29, 30, 3, 33, 34, 36, 37, 8, 39, 0, 4, 4, 43, 4, 46, 4, 8, 49, 0, 5, 5, 5, 54, 5, 57, 58, 59, 60, 6, 63, 6, 6, 66, 67, 69, 70, 7, 7, 73, 74, 76, 77, 78, 79, 80, 8, 8, 8, 8, 8, 87, 88, 8, 90, 9, 9, 93, 94, 9, 96, 97, 8, 99, 0, 20
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OFFSET
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1,2
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COMMENTS
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Define the "sieve" of a number k to be the list of divisors of k: {1, d_2, d_3, ..., k}.
Definition of sieving over the digits of k: Erase each digit 1 in the decimal expansion of k, then consolidate the remaining digits. Erase each digit d_2 in what remains from the previous step, then consolidate the remaining digits. Repeat the procedure with d_3, ..., largest divisor of k <= last consolidated remainder. What remains then becomes a term of the sequence. If there are no remaining digits after the procedure, this number disappears and is not a term.
Consolidation means the removal of all empty places at each step of the sieving process. Example: k = 1225; the divisors of k : {1,5,7,25,35,49,175,245,1225}; erasing all 1's in 1225 results in _225, which consolidates to 225; erasing all 5's in 225 results in 22_, which consolidates to 22; erasing all 7's in 22 results in 22. As there is no other divisor of 1225 <= 22 (and > 7) to sieve with, the result for k = 1225 after three sieving steps is 22.
The ratio c of disappearing numbers ("contraction"): for k in [1..99], c = 2/3; for k in [100..199], c = 3/20. What is c for k in [1,oo] ?
The number of sieving steps s(k) of the number k: 1 <= s(k) <= A000005(k).
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LINKS
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EXAMPLE
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k = 68; divisors of 68: {1,2,4,17,34,68}.
d_1 = 1, no occurrence of 1 in 68
d_2 = 2, no occurrence of 2 in 68
d_3 = 4, no occurrence of 4 in 68
d_4 = 17, no occurrence of 17 in 68
d_5 = 34, no occurrence of 34 in 68
d_6 = 68, 1 occurrence of 68 in 68, no remainder.
The number 68 disappears after 6 sieving steps and is not a member of the sequence.
k = 84; divisors of 84: {1,2,3,4,6,7,12,14,21,28,42,84}.
d_1 = 1, no occurrence of 1 in 84
d_2 = 2, no occurrence of 2 in 84
d_3 = 3, no occurrence of 3 in 84
d_4 = 4, 1 occurrence of 4 in 84, erase 4, remains 8
d_5 = 6, no occurrence of 6 in 8
d_6 = 7, no occurrence of 7 in 8
As there is no other divisor of 84 <= 8 (and > 7) to sieve with, the result for k = 84 after six sieving steps is 8. Number 8 is thus a member of the sequence.
k = 106; divisors of 106: {1,2,53,106}.
d_1 = 1, 1 occurrence of 1 in 106, erase 1, remains 06 which equals to 6
d_2 = 2, no occurrence of 2 in 6
As there is no other divisor of 106 <= 6 (and > 2) to sieve with, the result for k = 106 after two sieving steps is 6. Number 6 is thus a member of the sequence.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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