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A173336 Numbers k such that tau(phi(k)) = sigma(sopf(k)).

%I A173336 #23 Sep 08 2022 08:45:50
%S A173336 8,9,25,36,49,54,96,100,320,441,495,704,891,1029,1080,1089,1260,1331,
%T A173336 1386,1400,1617,1701,1750,1815,1848,1950,1960,2079,2541,2574,2704,
%U A173336 2850,2880,3000,3360,3430,3510,3861,4125,4275,4680,4704,4719,4800,5070,5096
%N A173336 Numbers k such that tau(phi(k)) = sigma(sopf(k)).
%C A173336 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 A173336 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 A173336 Amiram Eldar, <a href="/A173336/b173336.txt">Table of n, a(n) for n = 1..10000</a>
%H A173336 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 A173336 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 A173336 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler&#39;s_phi_function">Euler's totient function</a>
%F A173336 k such that A062821(k) = sigma(A008472(k)).
%e A173336 8 is in the sequence because phi(8) = 4, tau(4)=3, sopf(8)=2 and sigma(2) = 3 ;
%e A173336 9 is in the sequence because phi(9) = 6, tau(6)=4, sopf(9)=3 and sigma(3) = 4.
%p A173336 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 A173336 sopf[n_] := Plus @@ (First@# & /@ FactorInteger[n]); Select[Range[2, 5100], DivisorSigma[0,EulerPhi[#]] == DivisorSigma[1, sopf[#]] &] (* _Amiram Eldar_, Jul 09 2019 *)
%o A173336 (Magma) [m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+Divisors(&+PrimeDivisors(m))]; // _Marius A. Burtea_, Jul 10 2019
%o A173336 (PARI) isok(n) = (n>1) && numdiv(eulerphi(n)) == sigma(vecsum(factor(n)[, 1])); \\ _Michel Marcus_, Jul 10 2019
%Y A173336 Cf. A000005, A000010, A000203, A001157, A001158, A001160, A001065, A002192, A008472, A062821.
%K A173336 nonn
%O A173336 1,1
%A A173336 _Michel Lagneau_, Feb 16 2010
%E A173336 Corrected and edited by _Michel Lagneau_, Apr 25 2010
%E A173336 Edited by _D. S. McNeil_, Nov 20 2010