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A316412 Positive numbers m so that deletion of some or none but not all digits from m yields a noncomposite number. 0
1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 311, 317 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Subsequence of A068669. It is easy to see that these are the only terms from the said sequence that satisfy our definition; there are no more terms < 10000. If there is one >= 10000 then there would be one in [1000, 9999]. A contradiction hence the sequence is finite and full.

Also noncomposites m (in base 10) for which the concatenation of every subsequence of digits of m is noncomposite (in base 10). - David A. Corneth, Aug 08 2018

LINKS

Table of n, a(n) for n=1..20.

EXAMPLE

317 is a member since all its subsequences, i.e., 3, 1, 7, 31, 17, 37, 317, are noncomposite.

313 is not a member since one of its subsequences (33) is composite.

MATHEMATICA

Select[Range[10^3], AllTrue[FromDigits /@ Union@ Rest@ Subsets@ IntegerDigits@ #, ! CompositeQ@ # &] &] (* Michael De Vlieger, Aug 05 2018 *)

PROG

(C++)

#include <iostream>

#include <queue>

int main() {

    int upper = 1000;

    // 0->composite, 1->prime, 2->member of the sequence

    auto *nums = new int[upper];

    for (int i = 0; i < upper; i++)

        nums[i] = 1;

    nums[0] = nums[1] = 2;

    std::queue<int> in_progress;

    in_progress.push(1);

    for (int i = 2; i < upper; i++) {

        if (nums[i] == 0) continue;

        // is a prime

        in_progress.push(i);

        for (int j = i + i; j < upper; j += i) {

            nums[j] = 0;

        }

    }

    while (!in_progress.empty()) {

        int p = in_progress.front();

        in_progress.pop();

        int div = 1;

        bool valid = true;

        while (div <= p) {

            int del = (p / (div * 10)) * div + (p % div);

            if (nums[del] != 2) {

                valid = false;

                break;

            }

            div *= 10;

        }

        if (valid) {

            nums[p] = 2;

            std::cout << p << ", ";

        }

    }

}

CROSSREFS

Subsequence of A068669.

Cf. A008578.

Sequence in context: A190222 A012884 A068669 * A100553 A175584 A216823

Adjacent sequences:  A316409 A316410 A316411 * A316413 A316414 A316415

KEYWORD

base,easy,fini,full,nonn

AUTHOR

Matej Kripner, Aug 04 2018

STATUS

approved

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Last modified August 6 12:33 EDT 2020. Contains 336246 sequences. (Running on oeis4.)