A077658 Least composite number not congruent to 0 (modulo the first n primes) which contains its greatest proper divisor as a substring of itself, both in base two.
4, 55, 55, 91, 407, 493, 493, 893, 1189, 1189, 1643, 1681, 1681, 7597, 7597, 7597, 7597, 7597, 7597, 7597, 7979, 7979, 9167, 9167, 11227, 11227, 11227, 28757, 28757, 28757, 28757, 28757, 28757, 28757, 28757, 36349, 36349, 36349, 39917, 39917
Offset: 0
Examples
a(0)=4 since 4_d = 100_b and its largest proper divisor is 2_d = 10_b is a substring and 4 is not prime. a(2) = 55 since 55_d = 110111_b and its largest proper divisor is 11_d = 1011_b is a substring and 55 is not prime nor congruent to 0 (modulo either 2 or 3). a(4) = 407 since 407_d = 110010111_b and its largest proper divisor is 37_d = 100101_b is a substring and 407 is not prime nor congruent to 0 (modulo either 2, 3, 5, or 7).
Crossrefs
Cf. A063138.
Programs
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Mathematica
a = {}; k = 1; Do[p = Table[ Prime[i], {i, 1, n}]; While[ PrimeQ[k] || Sort[Mod[k, p]] [[1]] == 0 || StringPosition[ ToString[ FromDigits[ IntegerDigits[k, 2]]], ToString[ FromDigits[ IntegerDigits[ Divisors[k] [[ -2]], 2]] ]] == {}, k++ ]; a = Append[a, k], {n, 0, 100}]; a