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A352332
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Numbers k for which k = phi(k') + phi(k''), where phi is the Euler totient function (A000010), k' the arithmetic derivative of k (A003415) and k'' the second arithmetic derivative of k (A068346).
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0
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4, 260, 294, 740, 1460, 3140, 3860, 5540, 8420, 10820, 15140, 19940, 21860, 24020, 24260, 27620, 37460, 40340, 46820, 49460, 55940, 61220, 70340, 85460, 101540, 114020, 124340, 132740, 133220, 144260, 148340, 149540, 155060, 162020, 164420, 167060, 170420, 173540
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OFFSET
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1,1
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COMMENTS
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If p > 3 is at the intersection of A023221 and A005383, then m = 20*p is a term. Indeed, m' = (20*p)' = 24*p + 20 = 4*(6*p + 5), m'' = (4*(6*p + 5))'= 4*(6*p + 6) = 24*(p + 1), phi (m') + phi(m'') = phi (4*(6*p + 5)) + phi(24*(p + 1)) = 2*(6*p + 4)) + phi(48*(p + 1)/2) = 2*(6*p + 4)) + 16*(p - 1)/2) = 12*p + 8 + 8*p - 8 = 20*p = m.
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LINKS
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EXAMPLE
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phi(4') + phi(4'') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
phi(260') + phi(260'') = phi(332) + phi(336) = 164 + 96 = 260, so 260 is a term.
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MATHEMATICA
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d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[200000], CompositeQ[#] && EulerPhi[d[#]] + EulerPhi[d[d[#]]] == # &] (* Amiram Eldar, Apr 10 2022 *)
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PROG
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(Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; [n:n in [2..174000]|not IsPrime(n) and n-EulerPhi(Floor(f(n))) eq EulerPhi(Floor(f(Floor(f(n)))))];
(PARI)
ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = my(adk=ad(k)); !isprime(k) && (k == eulerphi(adk) + eulerphi(ad(adk))); \\ Michel Marcus, Apr 30 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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