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 A157037 Numbers with prime arithmetic derivative A003415. 20
 6, 10, 22, 30, 34, 42, 58, 66, 70, 78, 82, 105, 114, 118, 130, 142, 154, 165, 174, 182, 202, 214, 222, 231, 238, 246, 255, 273, 274, 282, 285, 286, 298, 310, 318, 345, 357, 358, 366, 370, 382, 385, 390, 394, 399, 418, 430, 434, 442, 454, 455, 465, 474, 478 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Only about 5% of these numbers have n' > n. For example, for n up to 10^6 we have 72754 prime n', with 69420 such that n' < n and 3334 such that n' > n. - Paolo P. Lava, May 20 2013, corrected by M. F. Hasler, Apr 07 2015 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 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 LINKS Antti Karttunen, Table of n, a(n) for n = 1..10001 (first 1000 terms from Reinhard Zumkeller) FORMULA A010051(A003415(a(n))) = 1; A068346(a(n)) = 1; A099306(a(n)) = 0. A003415(a(n)) = A328385(a(n)) = A241859(n); A327969(a(n)) = 3. - Antti Karttunen, Oct 19 2019 EXAMPLE A003415(42) = A003415(2*3*7) = 2*3+3*7+7*2 = 41 = A000040(13), therefore 42 is a term. MATHEMATICA 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 *) PROG (Haskell) a157037 n = a157037_list !! (n-1) a157037_list = filter ((== 1) . a010051' . a003415) [1..] -- Reinhard Zumkeller, Apr 08 2015 (PARI) A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1])); isA157037(n) = isprime(A003415(n)); \\ Antti Karttunen, Oct 19 2019 CROSSREFS Cf. A003415, A010051, A038554, A192082, A192189, A192190,  A327978, A328233, A328240, A328384, A328385. Cf. A189441 (primes produced by these numbers), A241859. Cf. A192192, A328239 (numbers whose 2nd and numbers whose 3rd arithmetic derivative is prime). Cf. A108605, A256673 (subsequences). Subsequence of following sequences: A005117, A099308, A235991, A328234 (A328393), A328244, A328321. Sequence in context: A083285 A082419 A102783 * A189992 A255746 A082917 Adjacent sequences:  A157034 A157035 A157036 * A157038 A157039 A157040 KEYWORD nonn AUTHOR Reinhard Zumkeller, Feb 22 2009 STATUS approved

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Last modified December 14 17:55 EST 2019. Contains 329979 sequences. (Running on oeis4.)