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A217305 Minimal natural number (in decimal representation) with n prime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime). 2
1, 2, 7, 13, 37, 88, 67, 192, 317, 932, 942, 1567, 4663, 4692, 8442, 23317, 23442, 36067, 102217, 114192, 180337, 192317, 511087, 901682, 582942, 2495443, 2555436, 2536067, 5289942, 12321061, 12680337, 12301692, 26461592, 61508461, 61508462, 63885918 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=5*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 5^j = (5^n - 1)/2 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-5) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-5 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be a prime number.

No term is divisible by 5.

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 0..53

FORMULA

a(n) > 5^floor(sqrt(8*n-7)-1)/2), for n>0.

a(n) <= (5^n - 1)/2, n>0.

EXAMPLE

a(1) = 2 = 2_5, since 2 is the least number with 1 prime substring in base-5 representation.

a(2) = 7 = 12_5, since 7 is the least number with 2 prime substrings in base-5 representation (2_5 and 12_5=7).

a(3) = 13 = 23_5, since 13 is the least number with 3 prime substrings in base-5 representation (2_5, 3_5, and 23_5).

a(4) = 37 = 122_5, since 37 is the least number with 4 prime substrings in base-5 representation (2 times 2_5, 12_5=7, and 122_5=37).

a(7) = 192 = 1232_5, since 192 is the least number with 7 prime substrings in base-5 representation (2 times 2_5, 3_5, 12_5=7, 23_5=13, 32_5=17, and 232_5=67).

CROSSREFS

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685, A035244, A079397, A213300-A213321, A217302-A217309.

Sequence in context: A330452 A086904 A291276 * A026555 A262727 A129592

Adjacent sequences:  A217302 A217303 A217304 * A217306 A217307 A217308

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Nov 22 2012

STATUS

approved

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Last modified November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)