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%I #58 Sep 15 2018 15:42:37
%S 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,
%T 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,
%U 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
%N Terms resulting from application of a divisor sieve to the digits of the decimal expansions of the positive integers.
%C Define the "sieve" of a number k to be the list of divisors of k: {1, d_2, d_3, ..., k}.
%C 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.
%C 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.
%C 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] ?
%C The number of sieving steps s(k) of the number k: 1 <= s(k) <= A000005(k).
%e k = 68; divisors of 68: {1,2,4,17,34,68}.
%e d_1 = 1, no occurrence of 1 in 68
%e d_2 = 2, no occurrence of 2 in 68
%e d_3 = 4, no occurrence of 4 in 68
%e d_4 = 17, no occurrence of 17 in 68
%e d_5 = 34, no occurrence of 34 in 68
%e d_6 = 68, 1 occurrence of 68 in 68, no remainder.
%e The number 68 disappears after 6 sieving steps and is not a member of the sequence.
%e k = 84; divisors of 84: {1,2,3,4,6,7,12,14,21,28,42,84}.
%e d_1 = 1, no occurrence of 1 in 84
%e d_2 = 2, no occurrence of 2 in 84
%e d_3 = 3, no occurrence of 3 in 84
%e d_4 = 4, 1 occurrence of 4 in 84, erase 4, remains 8
%e d_5 = 6, no occurrence of 6 in 8
%e d_6 = 7, no occurrence of 7 in 8
%e 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.
%e k = 106; divisors of 106: {1,2,53,106}.
%e d_1 = 1, 1 occurrence of 1 in 106, erase 1, remains 06 which equals to 6
%e d_2 = 2, no occurrence of 2 in 6
%e 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.
%Y Cf. A000005, A000027, A027750, A318708.
%K nonn,base
%O 1,2
%A _Ctibor O. Zizka_, Sep 10 2018
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