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Numbers k such that tau(phi(k)) = sigma(sopf(k)).
1

%I #23 Sep 08 2022 08:45:50

%S 8,9,25,36,49,54,96,100,320,441,495,704,891,1029,1080,1089,1260,1331,

%T 1386,1400,1617,1701,1750,1815,1848,1950,1960,2079,2541,2574,2704,

%U 2850,2880,3000,3360,3430,3510,3861,4125,4275,4680,4704,4719,4800,5070,5096

%N Numbers k such that tau(phi(k)) = sigma(sopf(k)).

%C tau(k) is the number of divisors of k (A000005); phi(k) is the Euler totient function (A000010); sigma(k) is the sum of divisors of k (A000203); and sopf(k) is the sum of the distinct primes dividing k without repetition (A008472).

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

%H Amiram Eldar, <a href="/A173336/b173336.txt">Table of n, a(n) for n = 1..10000</a>

%H P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1919), 75-113.

%H Wacław Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>, Elementary theory of numbers, Warszawa, 1964.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler&#39;s_phi_function">Euler's totient function</a>

%F k such that A062821(k) = sigma(A008472(k)).

%e 8 is in the sequence because phi(8) = 4, tau(4)=3, sopf(8)=2 and sigma(2) = 3 ;

%e 9 is in the sequence because phi(9) = 6, tau(6)=4, sopf(9)=3 and sigma(3) = 4.

%p with(numtheory): for n from 1 to 18000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = sigma(t2) then print (n): else fi : od :

%t sopf[n_] := Plus @@ (First@# & /@ FactorInteger[n]); Select[Range[2, 5100], DivisorSigma[0,EulerPhi[#]] == DivisorSigma[1, sopf[#]] &] (* _Amiram Eldar_, Jul 09 2019 *)

%o (Magma) [m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+Divisors(&+PrimeDivisors(m))]; // _Marius A. Burtea_, Jul 10 2019

%o (PARI) isok(n) = (n>1) && numdiv(eulerphi(n)) == sigma(vecsum(factor(n)[, 1])); \\ _Michel Marcus_, Jul 10 2019

%Y Cf. A000005, A000010, A000203, A001157, A001158, A001160, A001065, A002192, A008472, A062821.

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 16 2010

%E Corrected and edited by _Michel Lagneau_, Apr 25 2010

%E Edited by _D. S. McNeil_, Nov 20 2010