A217302 - OEIS

A217302

Minimal natural number (in decimal representation) with n prime substrings in binary representation (substrings with leading zeros are considered to be nonprime).

1, 2, 5, 7, 11, 15, 27, 23, 31, 55, 47, 63, 111, 95, 187, 127, 223, 191, 381, 255, 447, 503, 383, 511, 1015, 895, 767, 1023, 1533, 1791, 1535, 1919, 3039, 3069, 3067, 3839, 3967, 6079, 6139, 6135, 7679, 8063, 8159, 12159, 12271, 15359, 16127

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OFFSET

0,2

COMMENTS

The sequence is well-defined in that for each n the set of numbers with n prime substrings in binary representation is not empty. Proof: A000975(n+1) has exactly n prime substrings in binary representation (see A000975).

All terms with n > 1 are odd.

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 0..300

FORMULA

a(n) >= 2^ceiling(sqrt(8*n+1)-1)/2).

a(n) <= A000975(n+1).

a(n+1) <= 2*a(n)+1.

EXAMPLE

a(1) = 2 = 10_2, since 2 is the least number with 1 prime substring (=10_2) in binary representation.

a(2) = 5 = 101_2, since 5 is the least number with 2 prime substrings in binary representation (10_2 and 101_2).

a(4) = 11 = 1011_2, since 11 is the least number with 4 prime substrings in binary representation (10_2, 11_2, 101_2 and 1011_2).

a(8) = 31 = 11111_2, since 31 is the least number with 8 prime substrings in binary representation (4 times 11_2, 3 times 111_2, and 11111_2).

a(9) = 47 = 101111_2, since 47 is the least number with 9 prime substrings in binary representation (10_2, 3 times 11_2, 101_2, 2 times 111_2, 1011_2, and 10111_2).

CROSSREFS

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079397, A213300-A213321, A217303-A217309, A000975.

Sequence in context: A084935 A239072 A317242 * A062409 A342939 A089781

Adjacent sequences: A217299 A217300 A217301 * A217303 A217304 A217305

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Nov 22 2012

STATUS

approved