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A130704
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Palindromic primes whose squares are the sum of three consecutive primes.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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, ....
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LINKS
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FORMULA
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EXAMPLE
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7^2 = 49 = 13 + 17 + 19.
11^2 = 121 = 37 + 41 + 43.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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base,nonn,less
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AUTHOR
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STATUS
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approved
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