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A011254 Numbers k such that phi(k) + sigma(k) = 4*k. 4
23760, 59400, 153720, 4563000, 45326160, 113315400, 402831360, 731601000, 803685120, 865950624, 919501200, 1178491680, 3504597120, 3786686400, 6429564000, 14924714400, 25310621952, 26998616736, 53138687040, 86955675840, 513969369984, 1054373308800, 1868445408960 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If (sigma(m)-phi(m))/(4*m-sigma(m)-phi(m)) is a prime integer p not dividing m, then p*m is in the sequence. 135230346701100 is in the sequence and not divisible by 24. - Jens Kruse Andersen, Feb 17 2009

If k=80*m is in the sequence and gcd(m,10) = 1 then 200*m is also in the sequence. Proof: phi(200*m) + sigma(200*m) = phi(200)*phi(m) + sigma(200)*sigma(m) = 80*phi(m) + 465*sigma(k) = (5/2)*(32*phi(m) + 186*sigma(m)) = (5/2)*(phi(80)*phi(m) + sigma(80)*sigma(m)) = (5/2)*(phi(80*m) + sigma(80*m)) = (5/2)*(phi(k) + sigma(k)) = (5/2)*(4*k) = 5/2*(4*80*m) = 4*(200*m) so 200*m is in the sequence. - Farideh Firoozbakht, Mar 30 2009

REFERENCES

David Wells, Prime Numbers: The Most Mysterious Figures in Math, Hoboken, New Jersey, John Wiley & Sons (2005), p. 75.

Zhang Ming-Zhi (typescript submitted to Unsolved Problems section of Monthly, Oct 01 1996.

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..25 (terms < 5*10^12)

Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Kelley Harris, On the classification of integers n that divide phi(n)+sigma(n), J. Num. Theory 129 (2009) 2093-2110

EXAMPLE

phi(23760) + sigma(23760) = 5760 + 89280 = 4*23760, so 23760 is in the sequence.

MATHEMATICA

Select[Range[1000000], DivisorSigma[1, #] + EulerPhi[#] == 4 # &] (* David Nacin, Feb 28 2012 *)

PROG

(PARI) is(n)=eulerphi(n)+sigma(n)==4*n \\ Charles R Greathouse IV, Nov 27 2013

CROSSREFS

Cf. A000010, A000203, A011251, A011774, A015704.

Sequence in context: A179918 A168169 A031649 * A226002 A066234 A237691

Adjacent sequences:  A011251 A011252 A011253 * A011255 A011256 A011257

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Jud McCranie

1178491680 from Farideh Firoozbakht, Jan 31 2006

2 more terms from Jud McCranie, Jan 31 2006

24 divides all known terms of the sequence. If this is true for the next five terms then they are 6429564000, 14924714400, 25310621952, 26998616736 and 53138687040. - Farideh Firoozbakht, Mar 11 2006

More terms from Jens Kruse Andersen, Feb 17 2009

a(21) from Donovan Johnson, Feb 28 2012

a(22)-a(23) from Donovan Johnson, Apr 04 2012

STATUS

approved

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Last modified October 14 09:25 EDT 2019. Contains 327995 sequences. (Running on oeis4.)