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A173336
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Numbers k such that tau(phi(k)) = sigma(sopf(k)).
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1
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8, 9, 25, 36, 49, 54, 96, 100, 320, 441, 495, 704, 891, 1029, 1080, 1089, 1260, 1331, 1386, 1400, 1617, 1701, 1750, 1815, 1848, 1950, 1960, 2079, 2541, 2574, 2704, 2850, 2880, 3000, 3360, 3430, 3510, 3861, 4125, 4275, 4680, 4704, 4719, 4800, 5070, 5096
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OFFSET
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1,1
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COMMENTS
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tau(k) is the number of divisors of k (A000005); phi(k) is the Euler totient function (A000010); sigma(k) is the sum of divisors of k (A000203); and sopf(k) is the sum of the distinct primes dividing k without repetition (A008472).
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
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LINKS
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FORMULA
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EXAMPLE
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8 is in the sequence because phi(8) = 4, tau(4)=3, sopf(8)=2 and sigma(2) = 3 ;
9 is in the sequence because phi(9) = 6, tau(6)=4, sopf(9)=3 and sigma(3) = 4.
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MAPLE
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with(numtheory): for n from 1 to 18000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = sigma(t2) then print (n): else fi : od :
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MATHEMATICA
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sopf[n_] := Plus @@ (First@# & /@ FactorInteger[n]); Select[Range[2, 5100], DivisorSigma[0, EulerPhi[#]] == DivisorSigma[1, sopf[#]] &] (* Amiram Eldar, Jul 09 2019 *)
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PROG
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(Magma) [m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+Divisors(&+PrimeDivisors(m))]; // Marius A. Burtea, Jul 10 2019
(PARI) isok(n) = (n>1) && numdiv(eulerphi(n)) == sigma(vecsum(factor(n)[, 1])); \\ Michel Marcus, Jul 10 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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