A273906 Primes equal to the concatenation of two nonzero palindromic numbers.

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 199, 211, 223, 227, 229, 233, 277, 311, 331, 337, 433, 443, 449, 499, 557, 577, 599, 661, 677, 733, 773, 811, 877, 881, 883, 887, 911, 977, 991, 997, 1013, 1019, 1117, 1151, 1171, 1181

Offset: 1

Author

Giovanni Teofilatto, Jun 03 2016

Keywords

• nonn
• easy
• base

Comments

The only palindrome in this sequence below 10^9 is 11 (per request of Giovanni Teofilatto). A004022 is a subsequence. - David A. Corneth, Jun 10 2016

If we have a concatenation of two palindromes A = A', B = B' which is palindromic, concat(A,B) =: A.B = (A.B)' = B'.A' = B.A, then A*(10^LB-1) = B*(10^LA-1) (LX = length of X) <=> A*R(LB) = B*R(LA), where R(n) = (10^n-1)/9. To have A.B prime we also must have gcd(A,B) = 1, thus A | R(LA) and B | R(LB). Such numbers are listed in A249647 (not A014950), the only palindromes there are of the form 1...1, 3...3 or 9...9. Thus the only palindromic terms in this sequence A273906 are the repunit primes A004022. - M. F. Hasler, Jun 10 2016

Examples

			The prime 1013 is a term since 101 and 3 are palindromic.
The prime 101 is not a term, since it is not a concatenation of two nonzero palindromic numbers.
The prime 131 is not a term because it is not a concatenation of two nonzero palindromic numbers.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..25912
• David A. Corneth, PARI program

Crossrefs

Cf. A000040, A002113, A004022, A014950, A096489, A249647, A273906.

Programs

Mathematica

Take[#, 62] &@ Select[Sort@ Map[FromDigits@ Flatten@ IntegerDigits@ # &, Tuples[#, 2]], PrimeQ] &@ Select[Range[10^3], Reverse@ # == # &@ IntegerDigits@ # &] (* Michael De Vlieger, Jun 03 2016 *)
nxtPal[n_]:=With[{c=Join[{2},Flatten[Table[{10*10^d,11*10^d},{d,0,10}]]]},SelectFirst[n+c,PalindromeQ]]; Take[Join[{11},Select[ #[[1]]*10^IntegerLength[ #[[2]]]+#[[2]]&/@ Flatten[{#,Reverse[#]}&/@Subsets[Join[Range[8],NestList[nxtPal,9,100]],{2}],1],PrimeQ]//Union],60] (* Harvey P. Dale, Dec 08 2024 *)

PARI

\\ See program link from David A. Corneth, Jun 10 2016.

Formula

a(n) = A096489(n+1), n=1..21. - R. J. Mathar, Jun 12 2016. (This is a pure accident, I think, since A096489 is a finite sequence. - N. J. A. Sloane, Jun 12 2016)

Extensions

More terms from Michael De Vlieger, Jun 03 2016