A070158 Duplicate of A055940.
133, 403, 583, 713, 817, 2077, 2623, 2923, 4453, 4717, 5311, 5773, 7093, 7747, 9313
Offset: 1
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The prime p=47 is in this sequence since phi[p]/(sigma[p]-p) = p-1 is an integer, as is the case for any other prime. The composite n=403=13*31 is in this sequence, since the ratio phi(n)/(sigma[n]-n) =360/(1+13+31)=8 is an integer. The first few composites in this sequence are 133,403,583,713,... (A055940).
Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n); If[IntegerQ[s], Print[n]], {n, 2, 1000}]
Select[Range[2,300],IntegerQ[EulerPhi[#]/(DivisorSigma[1,#]-#)]&] (* Harvey P. Dale, Dec 25 2019 *)
for(n=2,999,eulerphi(n)%(sigma(n)-n) || print1(n",")) \\ M. F. Hasler, Dec 11 2007
In A062972, n=15: q = 8/8 = 1; n=101: q = 100/1 = 100. While integer quotient chowla(n)/phi(n) gives only 5 nonprime solutions below 20000000 (see A070037), here, the integer reciprocals, q = phi(n)/chowla(n) obtained with squared primes and with other composites. If n=p^2, q = p(p-1)/p = p-1. So for squared primes, the quotients give A006093.
Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n-1); If[IntegerQ[s], Print[n]], {n, 2, 100000}]
n=35: phi(35)=24, sigma(35)=1+5+7+35=48, chowla(35)=12, quotient=2
Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n-1); If[ !PrimeQ[n]&&!PrimeQ[Sqrt[n]]&&IntegerQ[s], Print[n]], {n, 2, 100000}]
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