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A359332 Numbers with arithmetic derivative which is a palindromic prime number (A002385).

%I A359332 #22 Mar 24 2025 06:02:28
%S A359332 6,10,114,130,174,182,222,231,255,273,286,298,357,358,455,574,622,870,
%T A359332 1015,1309,1335,1677,1695,12594,13630,13686,15258,18534,18654,19082,
%U A359332 19114,19522,19626,19922,19986,20998,21558,22178,22882,22930,23062,23262,23709,24338
%N A359332 Numbers with arithmetic derivative which is a palindromic prime number (A002385).
%C A359332 A subsequence of A157037.
%C A359332 If p and q, (p < q), are twin primes and q is a term in A002385, then m = 2*p is a term. Indeed, m' = (2*p)' = p + 2 = q, which is a palindromic prime number (A157037).
%e A359332 6' = 5 = A002385(3).
%e A359332 114' = 101 = A002385(6).
%p A359332 d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
%p A359332 q:= n-> (k-> isprime(k) and StringTools[IsPalindrome](""||k))(d(n)):
%p A359332 select(q, [$1..25000])[];  # _Alois P. Heinz_, Jan 29 2023
%t A359332 d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[25000], PrimeQ[p = d[#]] && PalindromeQ[p] &] (* _Amiram Eldar_, Jan 29 2023 *)
%o A359332 (Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>;
%o A359332 pal:=func<n|Intseq(n) eq Reverse(Intseq(n))>;
%o A359332 [p:p in [1..25000]|pal(Floor(f(p))) and IsPrime(Floor(f(p)))];
%Y A359332 Cf. A001097, A002113, A002385, A003415, A157037.
%K A359332 nonn,base
%O A359332 1,1
%A A359332 _Marius A. Burtea_, Jan 29 2023