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A055940
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Counterbalanced numbers: Composite numbers k such that phi(k)/(sigma(k)-k) is an integer.
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5
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133, 403, 583, 713, 817, 2077, 2623, 2923, 4453, 4717, 5311, 5773, 7093, 7747, 9313, 11023, 11581, 11653, 12877, 14353, 15553, 19303, 20803, 21409, 21733, 21971, 24307, 31169, 35033, 39283, 39337, 43873, 46297, 46357, 50573, 50879, 53863
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OFFSET
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1,1
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COMMENTS
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Banks and Luca (2007) showed that the number of terms <= x, N(x) <= x * exp(-((1/3)*(log(8))^(1/3) + o(1))*(log(x))^(1/3)*(log(log(x)))^(1/3)) as x -> infinity, and that under Dickson's conjecture this sequence is infinite, since for each positive integer m, if p = 5m + 1 and q = 20m + 13 are primes, then p*q is a term. - Amiram Eldar, Apr 13 2020
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LINKS
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EXAMPLE
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k = 133 = 7*19: phi(133)=108, sigma(133)-133 = 1+7+19 = 27, q = 4.
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MATHEMATICA
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Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n); If[ !PrimeQ[n]&&IntegerQ[s], Print[n]], {n, 2, 1000000}]
Select[Range[54000], CompositeQ[#]&&IntegerQ[EulerPhi[#]/(DivisorSigma[ 1, #]-#)]&] (* Harvey P. Dale, Nov 16 2021 *)
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PROG
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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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