A035244 Smallest number with exactly n prime substrings.

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

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

Cf. A035232, A079397.

Sequence in context: A285789 A090528 A094535 * A085822 A213321 A093301

Adjacent sequences: A035241 A035242 A035243 * A035245 A035246 A035247

Keyword

base,easy,nonn

Author

Erich Friedman

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