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%I #48 Mar 16 2024 11:18:51
%S 6,10,22,30,34,42,58,66,70,78,82,105,114,118,130,142,154,165,174,182,
%T 202,214,222,231,238,246,255,273,274,282,285,286,298,310,318,345,357,
%U 358,366,370,382,385,390,394,399,418,430,434,442,454,455,465,474,478
%N Numbers with prime arithmetic derivative A003415.
%C Equivalently, solutions to n'' = 1, since n' = 1 iff n is prime. Twice the lesser of the twin primes, 2*A001359 = A108605, are a subsequence. - _M. F. Hasler_, Apr 07 2015
%C All terms are squarefree, because if there would be a prime p whose square p^2 would divide n, then A003415(n) = (A003415(p^2) * (n/p^2)) + (p^2 * A003415(n/p^2)) = p*[(2 * (n/p^2)) + (p * A003415(n/p^2))], which certainly is not a prime. - _Antti Karttunen_, Oct 10 2019
%H Antti Karttunen, <a href="/A157037/b157037.txt">Table of n, a(n) for n = 1..10001</a> (first 1000 terms from Reinhard Zumkeller)
%F A010051(A003415(a(n))) = 1; A068346(a(n)) = 1; A099306(a(n)) = 0.
%F A003415(a(n)) = A328385(a(n)) = A241859(n); A327969(a(n)) = 3. - _Antti Karttunen_, Oct 19 2019
%e A003415(42) = A003415(2*3*7) = 2*3+3*7+7*2 = 41 = A000040(13), therefore 42 is a term.
%t dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Select[Range[500], dn[dn[#]] == 1 &] (* _T. D. Noe_, Mar 07 2013 *)
%o (Haskell)
%o a157037 n = a157037_list !! (n-1)
%o a157037_list = filter ((== 1) . a010051' . a003415) [1..]
%o -- _Reinhard Zumkeller_, Apr 08 2015
%o (PARI)
%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o isA157037(n) = isprime(A003415(n)); \\ _Antti Karttunen_, Oct 19 2019
%o (Python)
%o from itertools import count, islice
%o from sympy import isprime, factorint
%o def A157037_gen(): # generator of terms
%o return filter(lambda n:isprime(sum(n*e//p for p,e in factorint(n).items())), count(2))
%o A157037_list = list(islice(A157037_gen(),20)) # _Chai Wah Wu_, Jun 23 2022
%Y Cf. A003415, A010051, A038554, A192082, A192189, A192190, A327978, A328233, A328240, A328384, A328385.
%Y Cf. A189441 (primes produced by these numbers), A241859.
%Y Cf. A192192, A328239 (numbers whose 2nd and numbers whose 3rd arithmetic derivative is prime).
%Y Cf. A108605, A256673 (subsequences).
%Y Subsequence of following sequences: A005117, A099308, A235991, A328234 (A328393), A328244, A328321.
%K nonn
%O 1,1
%A _Reinhard Zumkeller_, Feb 22 2009
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