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A055465
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Composite numbers k for which phi(k) + sigma(k) is an integer multiple of the number of divisors of k.
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4
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4, 15, 21, 25, 30, 33, 35, 39, 45, 48, 49, 51, 55, 56, 57, 65, 69, 70, 77, 78, 81, 85, 87, 91, 93, 95, 99, 102, 105, 110, 111, 115, 119, 121, 123, 125, 126, 129, 133, 135, 140, 141, 143, 145, 147, 153, 155, 159, 161, 165, 168, 169, 174, 177, 180, 182, 183, 184
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OFFSET
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1,1
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COMMENTS
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Makowski proved that phi(k) + sigma(k) = k*d(k) if and only if k is a prime (see in Sivaramakrishnan,Chapter I, page 8, Theorem 3). In this more general case the right hand side is instead m*d(k), and this equation holds for all primes.
This sequence is infinite: if p == 1 (mod 3) is prime then p^2 is a term. - Amiram Eldar, Mar 25 2024
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REFERENCES
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R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Marcel Dekker, Inc., New York and Basel, 1989.
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LINKS
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C. A. Nicol, Problem E 1674, The American Mathematical Monthly, Vol. 71, No. 3 (1964), p. 317; Another characterization of prime number, Solutions to Problem E 1674 by Martin J. Cohen and J. A. Fridy, ibid., Vol. 72, No. 2 (1965), pp. 186-187.
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FORMULA
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Composite integer solutions of phi(x) + sigma(x) = m * d(x) or A000010(x) + A000203(x) = m * A000005(x), where m is an integer.
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EXAMPLE
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78 is a term since it has 8 divisors, phi(78) = 24, sigma(78) = 168, and 24 + 168 = 192 = 24 * 8.
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MATHEMATICA
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Select[Range[184], CompositeQ[#] && Divisible[(DivisorSigma[1, #] + EulerPhi[#]), DivisorSigma[0, #]] &] (* Jayanta Basu, Jul 12 2013 *)
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PROG
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(PARI) is(n)=my(f=factor(n)); (eulerphi(f)+sigma(f))%numdiv(f)==0 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Mar 01 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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