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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

2n = sigma(n) + phi(n) if and only if n is 1 or a prime.

If 7*2^n-1 is prime then m=2^(n+2)*3*(7*2^n-1) is in the sequence. Because phi(m)=2^(n+2)*(7*2^n-2); sigma(m)=7*2^(n+2)*(2^(n+3)-1) so phi(m)+sigma(m)=2^(n+2)*((7*2^n-2)+(7*2^(n+3)-7))=2^(n+2)* (63*2^(n+2)-9)=3*(2^(n+2)*3*(7*2^n-1))=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), 359-360.

Zhang Ming-Zhi (typescript submitted to Unsolved Problems section of Monthly, 96-01-10)

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], 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

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

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Last modified December 22 21:10 EST 2014. Contains 252372 sequences.