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A035244 Smallest number with exactly n prime substrings. 49
1, 2, 13, 23, 113, 137, 373, 1137, 1733, 1373, 11317, 11373, 13733, 31373, 113173, 131373, 137337, 337397, 1113173, 1137337, 1373373, 2337397, 3733797, 11373137, 11373379, 13733797, 37337397, 111373379, 123733739 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
No leading 0's allowed in substrings.
The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof by induction: '1' has 0 prime substrings and '2' has 1 prime substring. Let m be a number with n prime substrings. Then 10m+2 is a number with n+1 prime substrings (since m and 10m have identical prime substrings, and '2' is one additional prime substring, but 10m+2 cannot be prime). - Hieronymus Fischer, Aug 26 2012
LINKS
Hieronymus Fischer, Table of n, a(n) for n = 0..40
FORMULA
a(n) > 10^floor((sqrt(8*n-7)-1)/2) for n > 0. - Hieronymus Fischer, Jun 25 2012
Min_{k>=n} a(k) <= A079397(n-1), n > 0. - Hieronymus Fischer, Aug 26 2012
a(n+1) <= 10*a(n) + 2. - Hieronymus Fischer, Aug 26 2012
EXAMPLE
a(4)=113 since 3, 11, 13 and 113 are prime and no smaller number works.
MATHEMATICA
f[n_] := Block[{s = IntegerDigits[n], c = 0, d = {}}, l = Length[s]; t = Flatten[ Table[ Take[s, {i, j}], {i, 1, l}, {j, i, l}], 1]; k = l(l + 1)/2; While[k > 0, If[ t[[k]][[1]] != 0, d = Append[d, FromDigits[ t[[k]] ]]]; k-- ]; Count[ PrimeQ[d], True]]; a = Table[0, {25}]; Do[ b = f[n]; If[ a[[b + 1]] == 0, a[[b + 1]] = n], {n, 1, 15000000}]; a
CROSSREFS
Sequence in context: A285789 A090528 A094535 * A085822 A213321 A093301
KEYWORD
base,easy,nonn
AUTHOR
EXTENSIONS
Edited by Robert G. Wilson v, Feb 25 2003
a(25)-a(40) from Hieronymus Fischer, Jun 25 2012 and Aug 25 2012
STATUS
approved

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Last modified April 20 00:03 EDT 2024. Contains 371798 sequences. (Running on oeis4.)