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

8, 9, 25, 36, 49, 54, 96, 100, 320, 441, 495, 704, 891, 1029, 1080, 1089, 1260, 1331, 1386, 1400, 1617, 1701, 1750, 1815, 1848, 1950, 1960, 2079, 2541, 2574, 2704, 2850, 2880, 3000, 3360, 3430, 3510, 3861, 4125, 4275, 4680, 4704, 4719, 4800, 5070, 5096

Offset: 1

Author

Michel Lagneau, Feb 16 2010

Keywords

• nonn

Comments

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).

Examples

			8 is in the sequence because phi(8) = 4, tau(4)=3, sopf(8)=2 and sigma(2) = 3 ;
9 is in the sequence because phi(9) = 6, tau(6)=4, sopf(9)=3 and sigma(3) = 4.
		

References

• 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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
• Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.
• Wikipedia, Euler's totient function

Crossrefs

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

Programs

Magma

[m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+Divisors(&+PrimeDivisors(m))]; // Marius A. Burtea, Jul 10 2019

Maple

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 :

Mathematica

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

PARI

isok(n) = (n>1) && numdiv(eulerphi(n)) == sigma(vecsum(factor(n)[, 1])); \\ Michel Marcus, Jul 10 2019

Formula

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

Extensions

Corrected and edited by Michel Lagneau, Apr 25 2010

Edited by D. S. McNeil, Nov 20 2010