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%I #45 Oct 21 2021 15:46:59
%S 1,2,1,4,6,1,2,10,12,1,16,18,22,4,2,28,30,1,36,40,42,46,6,52,58,60,1,
%T 66,70,72,78,2,82,88,96,100,102,106,108,112,10,4,126,1,130,136,138,
%U 148,150,156,162,166,12,172,178,180,190,192,196,198,210,222,226,228,232
%N Let m be the n-th positive integer such that phi(m) is divisible by m - phi(m). Then a(n) = phi(m)/(m - phi(m)).
%C m is the n-th prime power larger than 1; i.e., m = A000961(n+1). Proof: If phi(m) is divisible by m-phi(m), then m is divisible by m-phi(m). Let k be the product of the distinct prime factors of m. Then phi(m)/m = phi(k)/k, so k/(k-phi(k)) = m/(m-phi(m)) is an integer. Thus k is divisible by k-phi(k) and k is squarefree. Let k-phi(k) = d and k/(k-phi(k)) = e; note that e>1 and GCD(d,e)=1. Thus d = k - phi(k) = d e - phi(d e) = d e - phi(d) phi(e) so d (e-1) = d e - d = phi(d) phi(e) <= phi(d) (e-1) and d <= phi(d). But this implies that d=1, so phi(k)=k-1 and k is prime. Hence m is a prime power. - _Dean Hickerson_, Aug 28 2001
%C For primes, quotient = (p - 1) / 1 = p - 1; for prime powers, p^a, a > 1: quotient = p^(a - 1)(p - 1) / p^(a - 1) = p - 1, so each p - 1 values occur infinitely often: a(n) + 1 = root of n-th prime power with positive exponent, i.e., A025473(n+1). - [Edited by] _Daniel Forgues_, May 08 2014
%C "LCM numeral system": a(n+1) is maximum digit for index n, n >= 0; a(-n) is maximum digit for index n, n < 0. - _Daniel Forgues_, May 03 2014
%H Harvey P. Dale, <a href="/A063872/b063872.txt">Table of n, a(n) for n = 1..1000</a>
%H OEIS Wiki, <a href="/wiki/LCM_numeral_system">LCM numeral system</a>
%F a(n) = A025473(n + 1) - 1. - _Bill McEachen_, Sep 11 2021
%t epd[n_]:=Module[{ep=EulerPhi[n]},If[Divisible[ep,n-ep],ep/(n-ep),Nothing]]; Array[epd,300,2] (* _Harvey P. Dale_, Dec 27 2020 *)
%o (PARI) M(n) = ispower(n, , &n); if (isprime(n), n, 1); \\ A014963
%o apply(x->x-1, select(isprime, apply(x->M(x+1), [1..260]))) \\ _Michel Marcus_, Sep 14 2021
%Y Cf. A000010, A051953, A007694, A000961, A054740, A049237, A025473.
%K easy,nonn
%O 1,2
%A _Labos Elemer_, Aug 27 2001
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