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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

%I

%S 1,2,3,5,7,11,13,17,23,31,37,53,71,73,113,131,137,173,311,317

%N Positive numbers m so that deletion of some or none but not all digits from m yields a noncomposite number.

%C 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.

%C 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

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

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

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

%o (C++)

%o #include <iostream>

%o #include <queue>

%o int main() {

%o int upper = 1000;

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

%o auto *nums = new int[upper];

%o for (int i = 0; i < upper; i++)

%o nums[i] = 1;

%o nums[0] = nums[1] = 2;

%o std::queue<int> in_progress;

%o in_progress.push(1);

%o for (int i = 2; i < upper; i++) {

%o if (nums[i] == 0) continue;

%o // is a prime

%o in_progress.push(i);

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

%o nums[j] = 0;

%o }

%o }

%o while (!in_progress.empty()) {

%o int p = in_progress.front();

%o in_progress.pop();

%o int div = 1;

%o bool valid = true;

%o while (div <= p) {

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

%o if (nums[del] != 2) {

%o valid = false;

%o break;

%o }

%o div *= 10;

%o }

%o if (valid) {

%o nums[p] = 2;

%o std::cout << p << ", ";

%o }

%o }

%o }

%Y Subsequence of A068669.

%Y Cf. A008578.

%K base,easy,fini,full,nonn

%O 1,2

%A _Matej Kripner_, Aug 04 2018

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Last modified December 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)