

A055465


Composite numbers for which sum of EulerPhi and DivisorSum is an integer multiple of the number of divisors.


1



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

1,1


COMMENTS

Makowski proved that phi(n) + sigma(n) = n*d(n) iff n is a prime (see in Sivaramakrishnan,Chapter I, page 8, Theorem 3). In this more general case the right hand side is instead k*d(n).


REFERENCES

Sivaramakrishnan,R.(1989):Classical Theory of Arithmetical Functions Marcel Dekker,Inc., New YorkBasel.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


FORMULA

Composite integer solutions of Phi[x]+Sigma[x] = kd[x] or A000203(n)+A000010(n) = k*A000005(n), where k is integer.


EXAMPLE

It is true for all primes and some composites. n = 78, 8 divisors, Sigma = 168, Phi = 24, 168+24 = 192 = 8*24


MATHEMATICA

Select[Range[184], ! PrimeQ[#] && Divisible[(DivisorSigma[1, #] + EulerPhi[#]), DivisorSigma[0, #]] &] (* Jayanta Basu, Jul 12 2013 *)


PROG

(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


CROSSREFS

Sequence in context: A100783 A170850 A297711 * A209870 A167293 A267769
Adjacent sequences: A055462 A055463 A055464 * A055466 A055467 A055468


KEYWORD

nonn


AUTHOR

Labos Elemer, Jun 27 2000


STATUS

approved



