A083660 Smallest nonnegative integer m such that the concatenation of the integers from n to 1 interspersed with those of m, in base 10, is prime.

1, 5, 14, 5, 5, 9, 1, 1, 29, 23, 28, 13, 46, 22, 18, 116, 35, 18, 155, 7, 81, 1, 139, 52, 262, 215, 56, 29, 11, 6, 256, 119, 381, 592, 67, 189, 116, 46, 5, 275, 139, 27, 101, 118, 96, 167, 196, 393, 275, 91, 146, 415, 193, 127, 85, 73, 6, 4, 50, 118, 1046, 362, 5, 431, 248, 180, 82, 230, 125

Offset: 2

Author

Farideh Firoozbakht, Jun 14 2003

Keywords

• nonn
• base

Comments

Firoozbakht's conjecture: there exists an a(n) for every n greater than 1 and it is less than n^2.

For n with one digit, the searched-for prime must have at least 2n - 1 digits in base 10.

Firoozbakht's conjecture holds true up to at least 100. With adequately coded commands, verification should not take longer than a minute. - Alonso del Arte, Dec 09 2009

Examples

			a(4) = 14 because the concatenation of the digits from 4 to 1 (that is, 4321) with 14 stuck between each of them is 4143142141, and that is a prime number. Similar concatenations with numbers less than 14 used in 14's place all give composite numbers.
		

Links

• C. Rivera, Puzzle 8 (www.primepuzzles.net).

Crossrefs

Cf. A032702.

Programs

Mathematica

(*In the absence of a base argument, the function leastGenPrimeByListingFNI assumes the base is 10. Minimum and maximum allowed base values are 2 and 36.*) leastGenPrimeByListingFNI[n_, b_: 10] := Module[{m = 0, p, flag = False}, While[Not[flag], m++; p = FromDigits[Flatten[{Table[{IntegerDigits[i, b], IntegerDigits[m, b]}, {i, n, 2, -1}], {1}}], b]; flag = PrimeQ[p]]; Return[m]]; Table[leastGenPrimeByListingFNI[n], {n, 2, 10}]

Extensions

Terms verified by Alonso del Arte, Dec 09 2009