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A340323
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Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).
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5
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1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54
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OFFSET
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1,2
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COMMENTS
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Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.
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LINKS
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FORMULA
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Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022
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EXAMPLE
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a(2^s) = 3 for all s>0.
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MAPLE
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f:= proc(n) local t;
mul((t[1]+1)*(t[1]-1)^(t[2]-1), t=ifactors(n)[2])
end proc:
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MATHEMATICA
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fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}];
Array[phi, 245]
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PROG
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(PARI) A340323(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, (f[i, 1]+1)*((f[i, 1]-1)^(f[i, 2]-1)))); \\ Antti Karttunen, Jan 06 2021
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CROSSREFS
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Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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