A317093 Terms resulting from application of a divisor sieve to the digits of the decimal expansions of the positive integers.

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

Offset: 1

Author

Ctibor O. Zizka, Sep 10 2018

Keywords

• nonn
• base

Comments

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).

Examples

			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.
		

Crossrefs

Cf. A000005, A000027, A027750, A318708.