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A011774
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Nonprimes k that divide sigma(k) + phi(k).
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7
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1, 312, 560, 588, 1400, 23760, 59400, 85632, 147492, 153720, 556160, 569328, 1590816, 2013216, 3343776, 4563000, 4695456, 9745728, 12558912, 22013952, 23336172, 30002960, 45326160, 52021242, 75007400, 113315400, 137617728, 153587720, 402831360, 699117024
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OFFSET
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1,2
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COMMENTS
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2*k = sigma(k) + phi(k) if and only if k is 1 or a prime.
If 7*2^j - 1 is prime then m = 2^(j+2)*3*(7*2^j - 1) is in the sequence. Because phi(m) = 2^(j+2)*(7*2^j - 2); sigma(m) = 7*2^(j+2)*(2^(j+3) - 1) so phi(m) + sigma(m) = 2^(j+2)*((7*2^j - 2) + (7*2^(j+3) - 7)) = 2^(j+2)* (63*2^(j+2) - 9) = 3*(2^(j+2)*3*(7*2^j - 1)) = 3*m, hence m is a term of A011251 and consequently m is a term of this sequence. A112729 gives such m's. - Farideh Firoozbakht, Dec 01 2005
Conjecture: For n > 1, a(n) is a Zumkeller number (A083207). Verified for all n in [2,63]. - Ivan N. Ianakiev, Jan 25 2023
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
Zhang Ming-Zhi, typescript submitted to Unsolved Problems section of Monthly, 96-01-10.
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LINKS
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EXAMPLE
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a(26) = 113315400: sigma = 426535200, phi = 26726400, quotient = 4.
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MATHEMATICA
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Do[If[Mod[DivisorSigma[1, n]+EulerPhi[n], n]==0, Print[n]], {n, 1, 2*10^7}]
Do[ If[ ! PrimeQ[n] && Mod[ DivisorSigma[1, n] + EulerPhi[n], n] == 0, Print[n] ], {n, 1, 10^8} ]
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PROG
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(PARI) sp(n)=my(f=factor(n)); n*prod(i=1, #f[, 1], 1-1/f[i, 1]) + prod(i=1, #f[, 1], (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1))
p=2; forprime(q=3, 1e6, for(n=p+1, q-1, if(sp(n)%n==0, print1(n", "))); p=q) \\ Charles R Greathouse IV, Mar 19 2012
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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