A230770 Numbers n such that sigma(n) + phi(n) is a composite number of the form p^k where p is a prime.

2, 4, 12, 15, 110, 121, 125, 511, 908, 2047, 31269, 58252, 180544, 2275680, 3776877, 4164717, 4835820, 8386433, 8388607, 32284479, 60333777, 82628532, 122016110, 174438012, 238609292, 513528686, 515718093, 919749786, 1043394771, 3851465145, 4264386607

Offset: 1

Author

Jahangeer Kholdi, Jan 07 2014

Keywords

• nonn
• nice

Comments

All semiprimes of the form 2^m-1 are in the sequence. Because if 2^m-1=p*q where p and q are prime then sigma(2^m-1)+phi(2^m-1)=(p+1)*(q+1)+(p-1)*(q-1)=2(p*q+1)=2^(m+1). 15, 511, 2047, 8388607 and 137438953471 are the first five such terms of the sequence.

Also if p=(2^m-5)/9 is prime then n=4*p is in the sequence. Because phi(n)+sigma(n)=9*p+5=2^m. 12, 908, 58952 and 77433143050453552574776799557806810784652 are the first four such terms of the sequence.

Let h(n)=sigma(n)+phi(n), except for n=4 and n=121 for all other known terms n of the sequence h(n) is of the form 2^m. Note that h(4)=3^2 and h(121)=3^5, what is the next term n of the sequence such that h(n) is odd?

Examples

			sigma(12)+phi(12)=sigma(15)+phi(15)=2^5,
sigma(180544)+phi(180544)=2^19.
		

Crossrefs

Cf. A000010, A000203, A038344, A065387.

Programs

Mathematica

h[n_]:=DivisorSigma[1,n]+EulerPhi[n];Do[a=h[n];If[Length[FactorInteger[a]] == 1 && !PrimeQ[a], Print[n]],{n, 123456789}]

PARI

is(n)=isprimepower(sigma(n)+eulerphi(n))>1 \\ Charles R Greathouse IV, Sep 04 2014

Extensions

a(24)-a(31) from Donovan Johnson, Feb 19 2014