A154523 Numbers k such that the smallest decimal digit of k equals the smallest decimal digit of prime(k).

11, 13, 18, 31, 41, 52, 62, 73, 80, 81, 110, 112, 113, 114, 115, 116, 121, 125, 128, 133, 135, 140, 141, 142, 152, 156, 157, 164, 167, 170, 180, 187, 188, 189, 191, 192, 193, 194, 195, 196, 198, 199, 211, 215, 216, 217, 218, 219, 221, 231, 241, 251, 261, 271

Offset: 1

Author

Juri-Stepan Gerasimov, Jan 11 2009

Keywords

• nonn
• base
• less

Comments

Natural density 1, since almost all numbers and almost all primes (thanks to the prime number theorem) contain the digit 0.

The first terms with smallest digit 1, 2, and 3 are listed in the Data section. The first with smallest digits 4, 5, and 6 are 644, 758, and 6666, respectively. While there are plenty of primes with no decimal digit smaller than 7 (see A106110), including many primes consisting only of the digits 8 and 9 (the 10th of which is prime(77777) = 989999; cf. A020472), it seems to me that finding a term in this sequence whose smallest digit is 7 or 8 should be a very difficult problem. - Jon E. Schoenfield, Feb 11 2019

Examples

			11 is a term because prime(11) =  31 (smallest digits: 1);
13 is a term because prime(13) =  41 (smallest digits: 1);
18 is a term because prime(18) =  61 (smallest digits: 1);
31 is a term because prime(31) = 127 (smallest digits: 1);
41 is a term because prime(41) = 179 (smallest digits: 1);
52 is a term because prime(52) = 239 (smallest digits: 2).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A020472, A106110.

Programs

Maple

A054054 := proc(n) min(op(convert(n,base,10)) ) ; end proc:
for n from 1 to 500 do if A054054(n) = A054054(ithprime(n)) then printf("%d,",n ) ; end if; end do: (End) # R. J. Mathar, May 05 2010

Mathematica

Transpose[Select[Table[{n,Prime[n]},{n,300}],Min[IntegerDigits[#[[1]]]] == Min[IntegerDigits[#[[2]]]]&]][[1]] (* Harvey P. Dale, Dec 18 2012 *)

Extensions

Corrected (221 inserted) by R. J. Mathar, May 05 2010

Definition clarified by Harvey P. Dale, Dec 18 2012