A318708 Terms resulting from application of a prime sieve to the digits of the decimal expansions of the positive integers.

1, 4, 6, 8, 9, 10, 1, 1, 14, 1, 16, 1, 18, 0, 1, 4, 6, 8, 9, 0, 1, 4, 6, 8, 9, 40, 4, 4, 44, 4, 46, 4, 48, 49, 0, 1, 4, 6, 8, 9, 60, 6, 6, 64, 6, 66, 6, 68, 69, 0, 1, 4, 6, 8, 9, 80, 81, 8, 8, 84, 8, 86, 8, 88, 90, 91, 9, 9, 94, 9, 96, 9, 98, 99, 100, 10, 10, 104, 10, 106, 10, 108, 0, 1, 4

Offset: 1

Author

Ctibor O. Zizka, Sep 01 2018

Keywords

• nonn
• base

Comments

Definition of sieving over the digits of k: Erase each digit 2 in the decimal expansion of k, then consolidate the remaining digits. Erase each digit 3 in what remains from the previous step, then consolidate the remaining digits. Repeat the procedure with 5, 7, ..., largest prime <= 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; erasing all 2's in 1225 results in 1__5, which consolidates to 15; erasing all 3's in 15 results in 15; erasing all 5's in 15 results in 1_, which consolidates to 1. So for k = 1225 the result after sieving is 1. Example: k = 10101; erasing all 2's, ..., 97's results in 10101; erasing 101's in 10101 results in ___01, which consolidates to the last consolidated remainder 01. As there is no prime <= 01 to sieve with, the result for k = 10101 after sieving is 1.

Largest number of a sieve <= last consolidated remainder.

This sequence sieve is: {primes}. There could be other sieve definitions: {binary numbers}, {even numbers}, {odd numbers}, {triangular numbers}, predefined set of numbers like {0,3,11,27}, etc.

Examples

			n = 113
p_1 = 2, no occurrence of 2 in 113
p_2 = 3, 1 occurrence of 3 in 113, erase 3, remains 11
p_3 = 5, no occurrence of 5 in 11
p_4 = 7, no occurrence of 7 in 11
p_5 = 11, 1 occurrence of 11 in 11, no remainder
number 113 disappears and is not a member of the seq.
n = 114
p_1 = 2, no occurrence of 2 in 114
p_2 = 3, no occurrence of 3 in 114
p_3 = 5, no occurrence of 5 in 114
p_4 = 7, no occurrence of 7 in 114
p_5 = 11, 1 occurrence of 11 in 114, erase 11, remains 4
number 4 is a member of the seq.
		

Crossrefs

Cf. A245770, A248831, A248804, A000027, A062115.

Programs

Mathematica

upto[n_] := Block[{s = ToString /@ Range[n]}, Do[s = StringReplace[s, ToString[p] -> ""], {p, Prime@ Range@ PrimePi@ n}]; ToExpression@ DeleteCases[s, ""]]; upto[115] (* Giovanni Resta, Sep 01 2018 *)