A345666 Composite numbers whose largest prime substring is greater than the record of all previous terms.

12, 15, 27, 110, 117, 119, 123, 129, 141, 143, 147, 153, 159, 161, 171, 183, 189, 297, 1010, 1030, 1070, 1090, 1113, 1127, 1131, 1137, 1139, 1149, 1157, 1167, 1173, 1179, 1191, 1197, 1199, 1211, 1227, 1233, 1239, 1241, 1251, 1257, 1263, 1269, 1271, 1281, 1293

Offset: 1

Author

Eyal Gruss, Jun 21 2021

Keywords

• nonn
• base

Examples

			a(1)=12 is the first composite containing a prime substring. Its largest prime substring is A345667(1)=2. It is the first nonzero composite index of A047814.
		

Crossrefs

Cf. A002808, A047814, A345667 (corresponding prime substrings).

Programs

Mathematica

lst={};max=m=0;Do[If[!PrimeQ@n,If[IntegerQ[s=Max@Select[FromDigits/@Subsequences@IntegerDigits@n,PrimeQ]],m=s]];If[m>max,max=m;AppendTo[lst,n]],{n,10000}];lst (* Giorgos Kalogeropoulos, Jun 25 2021 *)

Python

def trojan_composites(limit_maxval=None, limit_terms=None, verbose=True):
    from sympy import isprime
    num = 1
    best = 0
    found = []
    while (not limit_maxval or num <= limit_maxval) and (not limit_terms or len(found) < limit_terms):
        num += 1
        if not isprime(num):
            string = str(num)
            for length in range(len(string), len(str(best)), -1):
                candidate = max(filter(isprime, {int(string[i:i + length - 1]) for i in range(len(string) - length + 2)}), default=0)
                if candidate:
                    if candidate > best:
                        best = candidate
                        found.append(num)
                        if verbose:
                            print(num, end=', ', flush=True)
                break
    if verbose:
        print()
    return found
trojan_composites(limit_terms=7) #[12, 15, 27, 110, 117, 119, 123]