

A011774


Nonprimes n that divide sigma(n) + phi(n).


7



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

1,2


COMMENTS

2n = sigma(n) + phi(n) if and only if n is 1 or a prime.
If 7*2^n1 is prime then m=2^(n+2)*3*(7*2^n1) is in the sequence. Because phi(m)=2^(n+2)*(7*2^n2); sigma(m)=7*2^(n+2)*(2^(n+3)1) so phi(m)+sigma(m)=2^(n+2)*((7*2^n2)+(7*2^(n+3)7))=2^(n+2)* (63*2^(n+2)9)=3*(2^(n+2)*3*(7*2^n1))=3*m, hence m is a term of A011251 and consequently m is a term of A011774. A112729 gives such m's.  Farideh Firoozbakht, Dec 01 2005


REFERENCES

R. K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359360.
Zhang MingZhi (typescript submitted to Unsolved Problems section of Monthly, 960110)


LINKS

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..63 (terms < 10^13, first 53 terms from Donovan Johnson)
Eric Weisstein's World of Mathematics, Prime Number


EXAMPLE

a(26)=113315400: sigma=426535200 phi=26726400 quotient=4


MATHEMATICA

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} ]


PROG

(PARI) sp(n)=my(f=factor(n)); n*prod(i=1, #f[, 1], 11/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, q1, if(sp(n)%n==0, print1(n", "))); p=q) \\ Charles R Greathouse IV, Mar 19 2012


CROSSREFS

Cf. A065387, A011251, A011254, A055681, A001771, A112729.
Sequence in context: A139638 A112542 A238099 * A011251 A043360 A237551
Adjacent sequences: A011771 A011772 A011773 * A011775 A011776 A011777


KEYWORD

nonn,nice


AUTHOR

R. K. Guy


EXTENSIONS

More terms from David W. Wilson
Corrected by Labos Elemer, Feb 12 2004


STATUS

approved



