A097645 -id:A097645 - cp's OEIS frontend

A018784 Numbers n such that sigma(phi(n)) = n.

1, 3, 15, 28, 255, 744, 2418, 20440, 65535, 548856, 2835756, 4059264, 4451832, 10890040, 13192608, 23001132, 54949482, 110771178, 220174080, 445701354, 4294967295, 16331433888, 18377794080, 94951936080, 204721968000, 386940247200, 601662398400, 1433565580920

Offset: 1

David W. Wilson

nonn

The numbers 2^2^n-1 for n=0,1,...,5 are in the sequence because 2^2^n-1=(2^2^0+1)*(2^2^1+1)*(2^2^2+1)*...*(2^2^(n-1)+1); 2^2^k+1 for k=0,1,2,3 & 4 are primes (Fermat primes); sigma(2^k)=2^(k+1)-1 and phi is a multiplicative function. Hence if p is a known Fermat prime (p=2^2^n+1 for n=0,1,2,3 & 4) then p-2 is in the sequence, note that this is not true for unknown Fermat primes if they exist. - Farideh Firoozbakht, Aug 27 2004

Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8. See Notes p. 7.

Cf. A097645, A097646, A019434.

Select[Range[10^6], DivisorSigma[1, EulerPhi[#]] == # &] (* Amiram Eldar, Dec 10 2020 *)

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

sigma(A001229), sorted.

Wilson's search was complete only through a(19) = 50319360. Jud McCranie reports Jun 15 1998 that the terms through a(24) are certain.

a(26)-a(28) added. Verified sequence is complete through a(28) by Donovan Johnson, Jun 30 2012

A097646 Numbers n such that n = phi(phi(n) + sigma(n)).

Original entry on oeis.org

1, 2, 6, 10, 20, 22, 46, 48, 58, 82, 106, 166, 178, 180, 208, 226, 262, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 864, 886, 982, 1018, 1120, 1186, 1282, 1306, 1318, 1366, 1368, 1438, 1486, 1522, 1618, 1822, 1906, 2026, 2038, 2062, 2098, 2206

Offset: 1

Farideh Firoozbakht, Sep 08 2004

nonn

If n=2*p where p is a Sophie Germain odd prime, then n is in the sequence; the proof is obvious.

			22 is in the sequence because phi(22)=10, sigma(22)=36 and phi(10+36)=22.

K. D. Bajpai, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Glossary, Sophie Germain prime.

Cf. A077080, A097652, A005384, A097645, A018784.

[n: n in [1..2300] | n eq EulerPhi(EulerPhi(n) + DivisorSigma(1,n))]; // Vincenzo Librandi, Aug 22 2015

with(numtheory):K:=proc()local n,a,c;  c:=1; for n from 1 to 5000000 do;
a:=phi(phi(n)+ sigma(n));if  a=n  then lprint(c,n); c:=c+1; fi;od; end:K(); # K. D. Bajpai, Jul 18 2013

Do[If[n==EulerPhi[EulerPhi[n]+DivisorSigma[1, n]], Print[n]], {n, 2400}]
Select[Range[2500],EulerPhi[EulerPhi[#]+DivisorSigma[1,#]]==#&] (* Harvey P. Dale, Jul 06 2021 *)

is(n)=sigma(n=factor(n))==eulerphi(eulerphi(n)) \\ Charles R Greathouse IV, Nov 27 2013

cp's OEIS Frontend

A018784 Numbers n such that sigma(phi(n)) = n.

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