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A291932
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a(n) is the smallest k such that (n+1)*phi(k) = (n-1)*psi(k).
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2
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2, 3, 95, 5, 143, 7, 319, 323, 559, 11, 117317, 13, 1007, 899, 1919, 17, 201983, 19, 441283, 1763, 394697, 23, 4031, 5249, 2911, 3239, 23519, 29, 3599, 31, 1796647, 979801, 8159, 5459, 5183, 37, 1550047, 10763, 8639, 41, 2709037, 43, 10207, 9179, 101567, 47, 12218993, 9701, 13199, 10403, 4018073, 53
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OFFSET
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2,1
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COMMENTS
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Least k such that Product_{p|k} (p+1)/(p-1) = (n+1)/(n-1). As a result, all terms are squarefree. - Charles R Greathouse IV, Sep 06 2017
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LINKS
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FORMULA
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a(p) = p for all primes p.
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EXAMPLE
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a(4) = 95 = 5*19 because (psi(5*19) + phi(5*19)) / (psi(5*19) - phi(5*19)) = (6*20 + 4*18) / (6*20 - 4*18) = 4 and 95 is the least number with this property.
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MAPLE
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N:= 10^7: # to get all terms before the first with a(n) > N
M:= nextprime(N):
A:= Vector(M):
R:= proc(n) mul((i[1]+1)/(i[1]-1), i=ifactors(n)[2]) end proc:
for k from 2 to N do
r:= R(k);
n:= (r+1)/(r-1);
if n::integer and n <= M and A[n] = 0 then
A[n]:= k;
fi
od:
m:=min(select(t -> A[t]=0, [$2..M]))-1:
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MATHEMATICA
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psi[n_] := If[n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors@ n}]]; f[n_] := Block[{k = 1}, While[(n + 1)*EulerPhi[k] != (n - 1)*psi[k], k++]; k]; Array[f, 52, 2] (* Robert G. Wilson v, Sep 06 2017 *)
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PROG
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(PARI) a(n)=my(target=2/(n-1)+1, start=n, end=10*n, f); while(1, forfactored(k=start, end, f=k[2][, 1]; if(vecmax(k[2][, 2])==1 && prod(i=1, #f, 2/(f[i]-1)+1)==target, return(k[1]))); start=end+1; end*=2) \\ Charles R Greathouse IV, Sep 06 2017
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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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