A130704 Palindromic primes whose squares are the sum of three consecutive primes.

7, 11, 151, 191, 929, 10301, 14741, 15451, 76667, 98689, 1062601, 1153511, 1175711, 1215121, 1300031, 1317131, 1489841, 1597951, 3075703, 3127213, 3362633, 3441443, 7354537, 7472747, 7662667, 9127219, 9196919, 9451549, 9561659

Offset: 1

Author

Robert G. Wilson v, Jun 19 2007

Keywords

• base
• nonn
• less

Comments

The number of such palindromic primes less than 10^n: 1, 2, 5, 5, 10, 10, 30, 30, 141, 141, 843, 843, 5856, 5856, 42675, 42675, ....

Examples

			7^2 = 49 = 13 + 17 + 19.
11^2 = 121 = 37 + 41 + 43.
		

Links

• Robert G. Wilson v, Table of n, for n = 1..1000.

Programs

Mathematica

NextPalindrome[n_] := Block[{l = Floor[Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]]]] > FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[ idn, Ceiling[l/2]], Reverse[Take[idn, Floor[l/2]]]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[ idfhn], Drop[ Reverse[ IntegerDigits[ idfhn]], Mod[l, 2]]]]]]]];
PrevPrim[n_] := Block[{k = n - 1}, While[ ! PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ ! PrimeQ[k], k++ ]; k]; fQ[n_] := Block[{p, q, r, s}, q = PrevPrim[ Ceiling[n^2/3]]; p = PrevPrim@q; r = NextPrim[ Floor[n^2/3]]; s = NextPrim@r; n^2 == p + q + r || n^2 == q + r + s];
pd = 6; lst = {}; Do[pd = NextPalindrome@pd; If[ PrimeQ@pd && fQ@pd, AppendTo[lst, pd]], {n, 10^8}]; lst
Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[10^7]],3,1]),PalindromeQ[#]&&PrimeQ[#]&] (* The program generates the first 8 terms of the sequence. To generate more, increase the Range constant, but the program may take a long time to run. *) (* Harvey P. Dale, Jul 26 2023 *)

Formula

Intersection of A002385 and A034961.