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A157037 Numbers with prime arithmetic derivative A003415. 19
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 November 15 06:21 EST 2019. Contains 329144 sequences. (Running on oeis4.)