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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 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 #include 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 = nums = 2;     std::queue 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 October 24 09:29 EDT 2021. Contains 348220 sequences. (Running on oeis4.)