A307003 Numbers x such that phi(x) = Sum_{i=1..k} phi(x/p_i), where p_i are the k prime factors of x.

2, 12, 24, 48, 96, 192, 252, 384, 504, 756, 768, 1008, 1365, 1512, 1536, 1820, 2016, 2268, 3024, 3072, 3640, 4032, 4536, 6048, 6144, 6804, 7280, 8064, 9072, 12096, 12288, 13608, 14560, 16128, 18144, 20412, 24192, 24576, 27216, 29120, 32256, 36288, 40824, 48384

Offset

1,1

Comments

Terms of A007283(n) are part of the sequence for n > 1. If x = 3*2^n then we have phi(3*2^n) = phi(3)*phi(2^n) = 2*(2-1)*2^(n-1) = 2^n and phi(n/3)+phi(n/2) = phi(2^n)+phi(3*2^(n-1)) = 2^(n-1)+2*(2-1)*2^(n-2) = 2*2^(n-1) = 2^n.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Example

Prime factors of 1365 are 3, 5, 7, 13 and phi(1365/3) + phi(1365/5) + phi(1365/7) + phi(1365/13) = 288 + 144 + 96 + 48 = 576 = phi(1365).

Maple

with(numtheory): P:=proc(n) local k;
if phi(n)=add(phi(n/k), k=factorset(n)) then n; fi; end:
seq(P(i), i=1..48384);

Mathematica

Select[Range[2, 1000], EulerPhi[#] == Total@EulerPhi[#/FactorInteger[#][[;; , 1]]] &] (* Amiram Eldar, Mar 20 2019 *)

Prog

(PARI) isok(n) = my(f=factor(n)[, 1]); eulerphi(n) == sum(k=1, #f, eulerphi(n/f[k])); \\ Michel Marcus, Mar 19 2019

Crossrefs

Cf. A000010, A007283, A324710, A324711.

Sequence in context: A195015 A051781 A077562 * A146567 A176679 A278407

Adjacent sequences: A307000 A307001 A307002 * A307004 A307005 A307006

Keyword

nonn

Author

Paolo P. Lava, Mar 19 2019

Status

approved