login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Numbers m such that phi(sum of divisors of m) = lambda(sum of distinct primes dividing m).
0

%I #18 Jan 13 2025 13:15:09

%S 3,6,10,22,34,142,214,382,862,2302,5182,279934,944782,1572862,1990654,

%T 114791254,127401982,339738622,8153726974,21743271934,4696546738174,

%U 112717121716222,158329674399742,169075682574334,300578991243262

%N Numbers m such that phi(sum of divisors of m) = lambda(sum of distinct primes dividing m).

%C Or numbers m such that A000010(A000203(m)) = A002322(A008472(m)), where phi is the Euler totient function and lambda is Carmichael's function.

%C Properties of the sequence:

%C (1) for n > 1, it seems that a(n) = 2*A078883(n) = 2*(Lesser member p of a twin prime pair such that p+1 is 3-smooth).

%C (2) {a(n)} is included in {A282515(n)}.

%C (3) for n > 2, a(n)/2 is a prime number congruent to 5 mod 6.

%e 34 is in the sequence because A000010(A000203(34)) = A000010(54) = 18, and

%e A002322(A008472(34)) = A002322(19) = 18.

%t Select[Range[10^6], EulerPhi@ DivisorSigma[1, #] == CarmichaelLambda[Total@ FactorInteger[#][[All, 1]]] &]

%o (PARI)

%o lambda(n) = lcm(znstar(n)[2]); \\ after _Charles R Greathouse IV_ in A002322

%o sopf(n) = vecsum(factor(n)[,1])

%o isok(n) = eulerphi(sigma(n)) == lambda(sopf(n)) \\ _Indranil Ghosh_, Mar 22 2017

%Y Cf. A000010, A000203, A002322, A008472, A078883, A282515.

%K nonn,more

%O 1,1

%A _Michel Lagneau_, Mar 22 2017